How much do you think the game would change if, say, pokémon had six moves? Or two tabs of “attack” moves and “no damage” moves?

I suppose “quite a bit.”  I’m actually on the record as being against increasing the number of moves Pokémon can learn, largely because – contrary to what you might intuitively expect – I think it would reduce strategy and diversity in the game, not increase it.  I expect it would slant the game towards the Pokémon with the largest and most diverse movepools, and reduce the trade-offs and calculations that go into building a team that covers its most important weaknesses with its limited resources.  I also suspect you might see a fairly dramatic increase in the importance of self-buffing movesets (which could, say, include a set-up move and some way to heal while still having room for four attacks) and consequently also things that counter those strategies (Haze, Whirlwind, Unaware, hell, maybe people would even start using Punishment).  Choice Band/Specs/Scarf, by contrast, probably gets slightly worse.  I could be completely wrong about all of this though; it’s the sort of thing might be interesting to see playtested.