Aside from the excellent points people provided here, I have to add 2 things.
Effective, governmental or otherwise, control.
This is the biggest problem and the crux of the issue. If this is a normal setting with no magic that can aid the government then normal banditry is an actual issue as there no effective policing so treasure hunting is even worth as there is no chance to actually lose anything.
So point is before even giving a license a chance a TH, treasure hunter, has to ask: But why should I bother?
So in video games you can go to the nearest dungeon or portal to hell, get a lot of loot, walk to the nearest town and exchange that for a lot of gold.
In this model there is 0 incentive to get a license.
But contrast that with unlicensed hunting in a developed state.
The risk of doing that is much higher and the government can track and apprehend the person way more effectively.
So you have to figure out a compelling reason why a TH has to get a license.
I would suggest something like taxes or proof of wealth or something similar.
This is difficult in the medieval world but if your TH can't sell those gems he acquired from a dungeon in the nearest town without a license then a black market exists or they have to get licenses.
Colleges has to be justified, but make them fit.
If you want then make it. I'd go with a place where rich adventurer seeking nobles go to learn about treasure hunting and be trained under experts.
Mind you this has to be highly specialized and they have to offer something real and useful.
Consider a son of a lord. He grew up in wealth with access to a fencing master and riding instructor and tutors on all subjects. So why would he bother going to that college unless it has to offer something he simply can't find at home?
This also runs into: if I have enough wealth I can simply gather a large enough party of men at arms, basically knights in all but name, and get what I want.
So the college has to offer stuff not available to the wealthy enough to be able to afford it.
Magic or enchanting or specific dungeon related things might come to mind
High risk high reward
It can be a way to move people up the social ladder.
This is similar to a lot of armies. Finish a 15 years service contract and you are reward with a piece of land. Congrats you are now a landowner, we all know this model.
So maybe the college exists to do something like that. Perhaps A bunch of originally poor THs gathered and made the college so that even the poorest with noble hearts and steady arms can make their fortune.
So the college can offer the poor a chance to be rich if they pass the trails and get a license and work the dungeons. Or just continue living.
Now the length of the contract and rate becomes a matter of balance.
You want them to bring in a steady supply of money but not be something like 90/10 split otherwise this is just stealing.
You can even tie this is with more dangerous contract offering better TH ratio but at the risk of being very dangerous.
Probably the most effective method of insuring that the TH does not run away with the chest of gold they found in the dungeon is magic. As this is a thing.
Armies have effective methods of dealing with desertion. DEATH.
You can also link the two
So lets assume that the world holds enough dungeons so it is actually viable to get rich by raiding them. And so the government and college set up a system where only the licensed THs can legally own the stuff from the dungeons. And since you can only get the license after the college clears it, then you have to go there.
This seems like a corrupt system, politics is corruption basically, but you can make it somewhat better by actually teaching the people how they handle dungeons and not die and so on.
Maybe your world actually requires that. Like new dungeons constantly appear and maybe even offer a threat. And thus a living can be made from clearing those.
I also use dungeon for stand in or as an example for other stuff.
Replace it with Dwemer ruins or Orc parties or haunted places or whatever you feel like. I only offer ideas to expand the subject and you can pick and choose.