> it suggests theres like a finite number that remains constant and they're just guessing which one is up this year

Close! Influenza is recombinant, meaning different viruses can swap out sections of genome with each other. The huge majority of viruses don't do this, and influenza is the only one that's super infectious.

You know how strains are named stuff like H1N1 or H5N8? H and N refer to the most important surface proteins, and the numbers are the specific variety. Within each variety there are tons and tons of mutation versions. So if you have an H-mutant virus and an N-mutant virus that seem to be doing well on their own, they may combine to become even more infectious. If you have two H-mutants, they are much less likely to suddenly become a problem unless they can find an N-mutation that is more infectious or more dangerous.

Note that they can also recombine with animal (eg pig or bird) strains, not just humans. It's not that it jumps species so much that it can potentially combine two dangerous aspects- the H protein may be dangerous in all species, but the N protein may be holding it back. If it picks up a human-compatible N protein, suddenly we're in trouble. We know this is how a lot of the more recent big waves of influenza happened, like H1N1.

Influenza mutates faster than covid even without recombination, but covid is SO infectious and immunity is still not widespread enough to keep up with mutations that are relatively small. Influenza is battling against centuries of built-up immunity, so it needs to be really lucky to have a sudden change.