So, I have been doing some research on the "glitch" used here.
What the run does is set RAM value 0x121C from 0 to 10. When that value is nonzero, if Start is pressed then instead of pausing the game the player will superjump upwards (even in the air), if A+Select are pressed the game ignores collision and locks in the player's vertical movement (allowing some slow hovering), and if B+Select are pressed, the game will switch to the next stage.
0x121C is set to 10 when 0x121D is set to 10. 0x121D increments by 1 under a specific set of circumstances, which I have not
entirely investigated, but they include that gameplay is active (0x1164 = 1), that the player is on the ground (0x121F and 0x1222 are non-zero), some other requirements I'm not entirely sure of (this check is not run if the game is paused, for instance), B and Select are held and A is pressed (HRAM 0x0B bits 1 and 2 are set, and HRAM 0x0C bit 0 is set). So basically, be in game, hold B and Select, and press A 10 times while on the ground (jump 10 times).
The nature of these requirements (holding some buttons down and pressing another a certain amount of times), and the effects that result from it, very strongly convince me that this is a debug code left in the game, and not a glitch.
Aside from that, knowing reliably how to do the actions needed, this movie can be improved by 270 frames:
User movie #38986309710552176 (goes up to the start of the last stage).
