I think starting time when starting the new game is fine but might have a variable loading time? Alternatively time could start when the first frame of the game is visible, idk if the game accepts any inputs before that though.

how about at least a category that bans the weapon wheel jumps?

for the All Puzzles run, we use a bit of RNG manipulation with the written out solution in the guide https://www.speedrun.com/t7g/guide/p2vlc