That is a large part of it. Maybe that solving this problem leads us to the answer to other problems. If you could prove the twin prime conjecture, that means

That is a large part of it. Maybe that solving this problem leads us to the answer to other problems. If you could prove the twin prime conjecture, that means you have figured out something about prime numbers. If you found a pattern or formula for figuring out the distances between twin primes, that means you're able to identify certain primes. And maybe that formula becomes the basis for a formula for the distance between primes in general. And once you have that formula, you have a simple way to test for primes. And I'm not saying that is happening or will happen or is even possible, but that would be a reason to study proofs of the twin prime conjecture.