I would recommend watching this computer play the game that a fellow speedrunner RingRush made. I can't explain a lot of the glitches in a small post like this, so give it a watch if you could. It's pretty fascinating:
youtu.be/cdE1_u_AAnQ?...

Without glitches, I believe you can skip some more moves if you use sandcastle cheats to cheese some requirements, but with that I think you can skip only Wonder Wing, Wading Boots, and Speed Shoes.

(Maybe Beak Bomb and Shock Spring, would have to check).

The computer can only climb that slope in the lair with an extremely technical glitch. You then need the note door move, which yes is a move that Bottles teaches you to open note doors.

I believe from then on the required moves are the same as the computer.

A human (you or me), requires a bit more moves, and I'm not 100% sure how many moves you need to skip in Spiral Mountain. You would probably add Punch, but I'm pretty sure that's skippable with a precise trick.

Then Talon Trot would be required to climb that first slope in the lair.

Hey there, speedrunner here. You can actually skip a lot more moves than beak bomb, but it varies whether a human or computer is playing the game.

To my knowledge, the computer playing the game only needs High Jump, Flutter, Beak Bust, Eggs, and Flight to beat the game.