Overall you have right thought about the situation, but incorrect conclusion. So, basically, it is a right question and a very very good question which has to be asked more often.
The "amount of people needed to support all the technology" is a part of a bigger question, and more general question - how big have to be a colony to be self-sufficient.
The answer is different for different tech levels. It is different not only in terms of low tech level needs one person to make stone knives and axes, the number of people varies also and for higher technological levels, if we compare your current productions with 1950x, 1900x.
The need to live in space sets a minimum for a technological level which has to be with rolling wheels.
Elon Musk (SpaceX) projection goal for Mars colony is about 1 million people, I would say that 20 million would be better, wheels will roll easier.
So far similar to what you describe, but as long as we include the Belt and thus the space with microgravity, everything changes a bit.
One of the benefits of microgravity and space is the energy costs, it costs less effort and energy to generate it, it is much cheaper than on earth or on a planet like Mars.
The reason for that is simple, no gravity means that a 1 um aluminium foil with area 1 km2 will keep its form without any support. You make a dish from it - it stays a dish, you make an elephant it stays in elephant form. It does not rust, it does not oxidize. The only thing it gets some holes from micrometeorites. (not totally true, but it just oversimplified picture of the difference)
The situation is not achievable on any planet or moon, but in space in microgravity is perfectly fine.
The second factor of the cheap energy, it does not ends (solar energy), even 3 a.u. it can be considered as cheap and endless. It does not depend on day/night, clouds, winds, you do not need accumulators, you do not need to dig for it - it is constant and almost the same power 24/7/365.
And now what?
Let's see Intel Fab42
Area about 200'000 m2, and I guess 50m height is enough - so potentially the Fab can be fitted into volume of 0.01 km3
So one cubic km can contain 100's of such fab's and different fabrics/manufacturers for the technological loop to be self-sufficient.
A 1km3 construction in microgravity is not a lot, it is not a problem compared to Burj Khalifa 829m height in 1g gravity. The construction can be light weight as it basically has no stress from gravity, and it can be stiff only in places where it is needed(fabric in a container).
How many people, intro
Development - yes it needs people, but if you do not have people for development it is not a problem, you will just have the same technology each year, and they will say it worked for my grand- grand- grandfather, so it will work for me. There is no dependence on development, they do not have to get more from the same surface, because they can just grow and scale the production according to the needs.
But if you need development, then yes, it needs a lot of people(but there are some solutions to the problem), but 20 million at current technological level is enough when everything is organized a bit more effectively than it is now.
The efficiency of production - also not a factor if they do not try to compete with someone else, and make the things just for their own consumption(and survival). If there is only 10% of good things(chips whatever) which pass quality control - it is not a problem when energy is order's of magnitudes cheaper, the scrap goes in the recycler.
Energy - as energy is not a problem then it is not a problem to keep a big factory(same Fab42) up and running 24/7/365 producing million of chip day, and do that just for one human, for him to have ability just to replace his phone, once a year in case the phone is lost or broken.
Used or not the energy never ends, and you can't limit it, you can stop convert it into useful work but that's all. If you do not convert it to useful work it flies into space to aliens.
The main factor will be, yes, how many people are needed to keep the thing operational and how many are needed to repair the thing. (actually, it is the same stuff but...)
At the moment high tech factories have pretty good automation, the human labour is in repairing the machines and the robots and in checking and quality control.
When you do not care about humans who have to work somewhere (because you do not have those resources, those labourers are the initial problem - no people, no problem), about efficiency, about high percentage of passing quality control production, about energy for recycling etc - then in that case it makes sense to put a robot arm in a place where we have humans at the moment, even it if kinda costs more and is slower and not so accurate - you just put it because you do not have a human to operate, but you have an ocean of cheap energy to compensate for possible flaws or the cost of the solution and when the only desired property is to not need humans to operate the facility.
In those circumstances, I guarantee you most of the workplaces can be replaced at today's technological level.
Repairing - do not repair, just put a new machine, and send the old machine in recycling(if there is no technology for repair, which is relatively easy can be done).
So 10 belt miners will have 10 cubic km mining complex which includes all factories needed and a huge foil mirror to supply the thing with solar energy - and they will be self-sufficient 100%. They can't create the technological seed, but they will be perfectly capable of managing it.
I did not mention a lot of other factors, such as example - lab equipment which can produce the same 7nm chips or other things - usually it is not that big, it is kinda sophisticated at its level but not so much, it is not very efficient especially for mass production - but as long as we do not care about efficiency it is not a problem, automation of such labs is possible etc
So answer is No, they will not depend a long time from anything, as a first technological seed will be produced. And the technological seed will be less than 1 km3 and it will need less than 100 people to manage it.