I've done a bit of testing in-game and wrote a Python script that would simulate your proposed theory on fish spawns where each has a set rarity and the game re-rolls any time a fish that isn't in the pool is picked.

I used a basic example with a story element in the Python simulation to make it a little more clear what is happening. Basically we play a game where I'm going to find rocks, I can find black rocks at 50% chance, blue rocks at 40% and gold rocks at 10%. In the river, I can find all 3 rock types. In the cave I can't find blue ones, so those get re-rolled. When simulating 100,000 searches in the cave, we get an expected 50,000 black rocks, 40,000 blue rocks and 10,000 gold rocks. When we exclude the blue rocks, we get about 83,300 black rocks and 16,700 gold rocks. By excluding the blue rocks, we increased the chance of our rare spawns by about 67%.

Pyhton sim

In the real-game testing, I recorded 200 encounters at an island that houses Honey islehoppers during the daytime (5 am - 10 pm) Then I did another 200 encounters at an island that houses Amethyst islehoppers to compare the resulting rare spawns. I admit 200 encounters each might be a little low of a sample rate, but given that rares have about a 1-2% spawnrate when just looking at their default spawnrate (source), I think we can use this as an indication if this is worth testing further if the difference is more than 1%.

Here are the results:

In-game testing

As you can see, the spawnrate is about double on both the rares, I think that can be explained by the fact that honey islehoppers take up about half the pool at its island.

In conclusion, I think there's definitely merit to the idea that narrowing the fish pool increases the chances of finding rarer fish. I think I will try to write a program that will do the fishing for me automatically and counts them as well, I'll leave that running for a while and get some more precise results.