Make paleontology relevant to everyday life
The problem with paleontology and why it's always the awkward unwanted stepchild when it comes to getting funding is that paleontology has absolutely no broader relevance to modern life. Even compared to similar disciplines such as the taxonomy or ecology of living organisms or history in general. Other evolutionary biologists like Richard Dawkins have even gone on record declaring their contempt for paleontology and how superfluous it is, saying things like "the evidence for evolution would be entirely secure, even if not a single corpse had ever fossilized". Or the elder Alvarez calling paleontologists "stamp collectors" and "not very good scientists" (though that one I will dispute because 1) paleontology has become more relevant since Alvarez' day and 2) he was a huge jerk who said things like this without doing any research and often made these half-baked, poorly-researched statements. Though his argument that a physicist managed to make paleontology more relevant than actual paleontologists still stands.
Compared to the study of living organisms paleontology comes up short because studies of the ecology and taxonomy of living animals can be applied to conservation efforts, controlling invasive species, using living animals as barometers to determine the health of an environment, pest control, etc. Paleontology doesn't do any of those things. And society has already shown how little it values biological taxonomists.
Paleontology is a historical science. It's basically chronicling the history of the Earth and all the things that live upon it. One might say "well, knowing where one came from is important, as well as what did and did not work in the past". That's true. But the problem is that nobody ever starved to death from not knowing their history. It's intellectually enriching, a nice bonus, but otherwise a superfluous luxury. Compare that to human history, which does have more relevance because it is about human behaviors, and therefore the policies and decisions that worked in the past can be more easily extrapolated to the present day. Those who do not learn from history and doomed to repeat it, and such.
There is no lesson that can be broadly applicable to social policy about how there was a massive volcanic event 225 million years ago at the end of the Triassic or that there was an inland sea in Kansas in the middle of the Cretaceous. No medical advances have ever been made by the discovery of Lucy and Ardi (and other hominins) and piecing together when and how human bipedalism originated. What knowledge can be applied to the present day (e.g., conservation efforts for Burramys finding out that the animal is actually more adaptive than we thought based on its fossil record, or "maybe we should be worrying about meteors") are rare exceptions that prove the rule.
Even Jack Horner's suggestion of using developmental biology to rebuild pseudo-dinosaurs from chickens has been laughed at for having no broader applications. Just because you've engineered a chicken to have teeth and claws doesn't mean it's going to know how to use them, or that it's behavior will give you any indication on how dinosaurs behaved. It's still going to act like a chicken. Even the genes or developmental signals you tweak to cause it to turn into a pseudo-dinosaur may not be the same ones that actual dinosaurs used when birds first evolved. Talking with colleagues we concluded the best use for such an animal would be to try to sell it as a curiosity to rich patrons for research money.
As someone who works in paleontology, you should see the kinds of mental gymnastics people do during grant season to try and justify how their research is relevant to the modern day. The big one nowadays is climate change (no, knowing how fast a certain mountain range was thrust up does not tell you how species will adapt to changing climates and anthropogenic habitat destruction).
People often say the fossil fuel industry makes paleontology relevant, but there's a saying in paleontology: "no oil company cares about your thesis on spinosaur paleoecology". Fossil fuel companies only care about a select group of paleontologists (people who study stratigraphy or certain microfossils like conodonts and forams), and even then you basically can't get a job as an oil geologist anymore because what used to require detailed geological knowledge and ferreting conodonts out of rock is now done by machines.
These features are why paleontology is almost always the first science on the chopping block at any major university. And a lot of scientific fields seem to thing the only thing paleontology is good for is getting people interested in science and bringing people into natural history museums.
What's the point of this long, seemingly nonsensical rant? In order to make paleontology the largest/most profitable industry in your setting it is necessary to understand why it is not so in our timeline. If you want to make paleontology the largest/most profitable industry in real life, you need to make it so that it has some direct application to real life that makes it relevant. Relevant enough that it doesn't just benefit humanity when funded (and therefore could be passed off as a luxury), but there have to be actual reasons why cutting funding would be bad. Look at other sciences. If medicine research funding gets cuts we don't have the resources to come up with new vaccines and cures. If engineering funding gets cut countries lose opportunities to make technological advancements. If ecology funding gets cut there are invasive species out there that we spend literal billions each year to contain that will become unleashed. If paleontology gets cut nothing happens. Indeed, this is what happens in IRL museums, the curators get fired and the collections get put in storage, the board of directors only cares about the dinosaur fossils on display.
Some people have brought up ancient technology, and I agree with them. Basically if you're in a technological arms race and how far ahead you are is dependent on what you dig up, then the people who do the digging are your lifeline. Except that would be archaeology or xeno-archaeology instead of paleontology. Paleontology focuses on non-sentient organisms.
Bringing things back to life is another option but even Michael Crichton pointed out there is no money to be made for bringing dinosaurs back to life. The whole reason he had Jurassic Park set in a dinosaur theme park was he couldn't think of a way that someone would want to clone dinosaurs and not go bankrupt in the process beyond "rich idiot decides to clone dinosaurs and make a zoo for them".
If they got out and things went full Dino Crisis you might have a reason for an adult knowing more than an eight-year-old about Tyrannosaurus rex. But even then it's technically interest in contemporary (if resurrected) animals that requires no knowledge of their fossil history. Field biologists would rapidly outpace paleontologists in their knowledge of these creatures due to accurate, first-hand experience of raptors trying to eat their faces.
Time travel is a good option. There you kind of have to know the landscape in order to know where things are, how to survive, and how to not get eaten by a T. rex. The downside is that from a relativity point of view it's more looking at contemporary animals than digging up bones, and what would happen really quickly is that scientists would stop digging up bones because you can get more by studying the living, breathing thing. One paper on a flesh-and-blood tyrannosaur would be worth more to science than 100 years of painstakingly pulling information from its fossil bones. Paleontology is like a roadmap to the past, only the map is torn, faded, written in a dead language, and somebody scribbled on it with magic marker.