Imagine a world where you can talk to anyone at all at any time, and telepathically transmit any amount of information to any select group of people. Why would a world like this ever get to the invention of radio, much less valuing it and transmitting?
Basically, in other words, what other uses could they find for radio?
Radio direction finding. (RDF) Cooking. (Microwave) Radars, sensor networks for non telepathic infrastructure (radio bouys for example), snooping on people covertly. (telepathy doesn't tell you what they are doing, only what they are thinking) Electronic harassment. Measurement equipment for meturlurgy and other material sciences. Emergency locators would still be desirable, because even if you can connect to somebody, that doesn't mean they know where they are, or are able to tell you. Astronomy (radio telescopes)
If they have machines like computers that perform tasks, these may not be able to perform telepathy as they can. Then they would sometimes need a way for computers to communicate wirelessly with each other. Radio isn't usually an ideal way for computers to do this, but it is one way they could have discovered radio and developed communication protocols around it.
The answer comes from the question, specifically select group of people. Radio has two purposes:
Two-way communication between select groups of people. This ranges from individual-to-individual communication on up to shared communication both structured - e.g., emergency responders, air-traffic control - and unstructured - e.g., CB Radio. The telepathic method would work great for individual communication and plausibly well for small groups - you just need a way to identify the group.
Broadcasting. Radio (and over-the-air TV) lets anyone, anywhere within range, receive broadcasts where the sender has no advance knowledge of who is receiving the broadcast. Telepathy doesn't (in my understanding of your world) work that way - the sender needs to (a) have a mental image of the recipients(s) and (b) the telepathic channel is always 2-way.
If I look at a stadium of people (too large for voice communication without amplification) I can instantly telepathically send to them, but I have to be prepared for the mental pressure of all of them (or even a small percentage) sending to me at the same time.
All the more so, if I want to communicate with anyone willing to listen, anywhere within hundreds of miles - including people I have never seen or been introduced to - I can't open the telepathic link, and if I can get past that (send me your picture and address?) then I have the fear of thousands of people trying to talk at one time.
Even worse if I am saying something inflammatory - the opposition would try to get in on the conversation and "mess with my head".
In addition, as @JBH pointed out, radio also allows nonverbal communication, particularly music.
So radio as mass media is the key use. Development would have to take a slightly different path from the real world as much of the initial development was as a wireless telegraph - replacement for a point-to-point wired connection. But it could definitely be done and would definitely be useful.
Machine communication? As in the real world, this will come later, possibly much later. While there are some very basic methods (e.g., radio controlled toy cars) that can be done without computers, the vast majority of what we think of as radio-based machine communications relies on computers. And not just any computers, integrated circuit based microcomputers. This requires advances in a number of different fields, far beyond what you need to have a big vacuum tube transmitter and crystal receivers, which is all you need to get radio functioning as mass media.
Radio is part of the electromagnetic (EM) spectrum. EM radiation includes radio, infrared, visible light, x-rays, gamma rays, etc. As they seek to understand the natural world, scientists will classify and quantify the portion of the EM spectrum that we call radio.
Any species that discovers radio will use it - they just might not use it the same way. Some useful applications include:
You're assuming basically godlike telepathic abilities
And that's unrealistic. Can you imagine the problem of having 7 billion people broadcasting to 7 billion people 24/7? That's unrealistic in the extreme. A 7,000 mile range is unrealistic, too.
Which means your people have the same need for radio as everyone else
What they don't need (possibly) are megaphones, microphones, and other near-space sound amplification for personal communication.
Which brings up another point...
Telepathy may obviate walkie-talkies, CB radios, and cell phones, but not radio
Because people love to listen to music! I suppose you could have one telepath listen to a lovely vinyl record and simultaneously broadcast what he/she is hearing, but that's a burden... that person would welcome the freedom radio would bring (if they could do it at all).
And then there's the need for computers to talk...
Because your computers aren't telepathic and the computer controlling the assembly robot needs to talk to that robot. Wires are so old-fashioned! Wireless machine-to-machine communication would by itself be a driver for radio. Could you imagine a telepath-based OnStarTM system? Hello, have you recently had an accident? Could you tell me where you are please? Hello? I think they're unconcious. Hello? Wait, is anyone even in the car? What do you mean you're going to the ba... Oh... sorry.
If you want to make this telepathy scientifically plausible -- make it via radio waves. On this planet beings have evolved which are able to communicate via radio waves. This is not completely implausible since on our planet we have things like sharks sensing electrical currents and birds navigating by sensing the magnetic field. Of course, the beings that have evolved this ability to communicate via radio don't originally know that they are doing this -- anymore then we knew much about how our own brain and nervous system worked until very recently (and even now, there is much to learn). Perhaps it is their biologists rather than their physicists who discover radio. They might think of it as "artificial telepathy" much the way that we think of computers as "artificial intelligence". Just like radio in our history was fairly unreliable for the first several decades, artificial telepathy would seem like a fairly poor substitute for natural telepathy at first (highly evolved natural radio communication is likely to have evolved things like error detecting codes and things like frequency hopping to resist jamming by predators). But just as AI is finally becoming competitive with humans in many areas, sooner or later artificial telepathy (radio) would start to outstrip natural telepathy in things like bandwidth. An intelligent species who all of a sudden understand the underlying physics of how they communicate could doubtless do much with that knowledge.
Billions people all capable of making you listen to whatever suits their whimsy is an absolutely terrible environment to live in. If even just a dozen abuse that power it's a miserable time for all; but more realistically the abusers will number in the thousands or even millions.
Technology that is capable of blocking, or at least limiting, telepathy is going to be developed pretty early in your civilization's history!
So, your civilization will still have the need for forms of communication that can happen without removing the block.
The answer can be pretty easily derived by looking at what issues are likely to be present that alternative communications methods would solve. Note that if you end up in a situation where verbal (or visual) communication is superior to this telepathy, you will still end up using radio for long-range communications, as it's the easiest means of long distance transmission of visual or verbal communications).
Looking at this as objectively as possible, given a very pessimistic interpretation of your description, you have a vast majority of the same issues here that you do with radio (interference, non-selectivity of transmissions, easily intercepted or blocked, shared media requiring regulation, etc), except all of those are much worse, because you can turn off a radio, but can't turn off your mind. In a case like this, it's very likely that people will either quickly find a way to block telepathy and then use some other form of communications, or, more likely, develop into a slightly more restricted form of collective consciousness akin to the Borg (and thus not need radio for communications at all).
Given a very optimistic interpretation though, you have a system very similar to a computer network as viewed from the application layer, and thus not many significant issues, and as a result radio would likely never develop as a means of direct communication, but would probably end up being developed as a means of broadcast communication, and possibly for things like music.
In both cases, it's likely to be developed for non-communications purposes such as radar, radio astronomy, and electronic communications.
It's probably worth noting that that pessimistic interpretation is the more likely one if this telepathy evolved before the development of sentience (coordinating a pack as a single unit is a huge advantage when hunting), while the optimistic interpretation is much more likely to be the result of an ability that was developed intentionally by an intelligent species (either through self modification, or through some form of external meddling), and something between the two is the most likely case if they developed it after developing sentience, but it originated naturally.
Scientific curiosity. All it takes is an eccentric mind to create something, there are many things invented that have no pressing need.
It seems to me that why or how he created it could be explained in many ways, perhaps he was trying to communicate with some other world or realm.
I was also thinking you could incorporate the eccentric into the story somehow later, make finding them a quest, or use as a mentor figure because of some deep mystical understanding.
If telepathy is only possible when you know who is on the other end of the connection (and it likely is because otherwise prank "calls" are going to be a problem), anonymity could be a large driving factor. A person could pick up a walkie-talkie from a dead drop and they wouldn't know who's on the other end. Very valuable if you're a member of the resistance.
Traditional vetting would eventually lead to using intermediaries to telepathically communicate (and direct telepathy for highly trusted people) but it would still retain value for recruiting new members.
Security might be lower than optimal for such a fledgling system, but if radio is a fairly esoteric means of communication it might be considered obscure enough that no one will figure out how to snoop. Or the resistance may simply be desperate enough. Or it could just be that the foreign state sponsor of the resistance has actually made the radios secure.
The simplest answer and probably the answer you want to hear is:
They want to be heard
They would discover it for the same reason we did - because it's there in nature, pretty much everywhere in the universe (for practical purposes), and is possibly an inevitable discovery once you get exploration of basic physics (17 - 19th century equivalent could probably make it a very likely/almost inevitable trajectory).
They would make use of it for almost identical reasons to the uses we have, assuming similar technological directions. Telepathy wouldn't change it much:
In truth: we do not invent "radio waves", it is as stupid as to say that we invent the light, since it is the same thing.
First thing: the radio waves exist whatever happens, except to create a universe without photon, but in this case, nothing is visible, the eyes have no function.
Second, even with telepathy, there is a maximum speed of information transfer in the universe, which is a constant that has been wrongly named: speed of light.
So radio waves go at this speed in a vacuum, in other words, nothing can be faster.
If you want to create a universe that violates Einstein's relativity, I wish you good luck on the coherence ;)
