I don't think it's necessarily the variable jump - even when holding the jump button for the full time, the increase in gravity feels a little too high. I think you might be overcorrecting for "floatiness" a little and could stand to reign it in a bit. That said... going back to the game, yeah a lot of the awkwardness is coming from how you've implemented the variable jump. Resetting the y velocity to 0 is just flat out not a good way to implement variable jump - it completely disrupts any sense of a jump arc and just feels incredibly unintuitive. My preference is to cut the velocity in half on the key release, but a lot of games do have the gravity scale increase happen on button release instead so since you already have that mechanic it might be worth using here.
Visuals are largely just not my taste I think, seems that some other people enjoyed them. I do understand that the stretched out sprites are a time constraints thing, but they stood out as particularly awkward looking to me.
For controls I'd probably prefer shift to teleport, yeah, though ideally they'd be freely rebindable (I know unity can make this a pain though so it's understandable that it's not an option here). I unfortunately don't have access to a controller so I can't judge how the controls are there, I'd imagine they're probably better.