You can see that Function Overload is still active in the clip when the third jump is done by Kai's scarf being bright green and the flashing red outline around him. If you still don't believe me you can try it yourself. Equip Air Jumper and Function Overload, go into any stage, activate Function Overload, and jump three times. You can keep doing it until the effect wears off.

Function Overload also grants an extra jump. If you combine both Air Jumper and Function Overload you get access to three jumps. With both active you can also endlessly jump cancel dive kicks, allowing you to stay in the air and keep gaining height until the effect wears off.

this is a very good decision thank you for actually really legitimately making this decision, glad this is definitely not a joke

Wouldn't it be best to use Seed Racer for runs since all upgrades behave as unlocked regardless of your main game progress? You aren't required to input a seed to use the mode, and you could just require people to show the menu screen for "seedless" runs to prove they did not input a seed.

Me and a couple others actually ran 0 star in Portal Mario 64 pretty competitively for a little while.

My first uploaded run

My current PB

Current WR

In that WR video you can also see that we use a save state to skip the intro and start the timer at 53.23 (when mario lands on the ground in the intro) so we don't have to sit through the intro every time we reset. 0 Star (and probably the other categories) are very reset heavy runs due to how buggy the portals are.