gv-usb2 is a popular choice, its like 40€. but if even thats too much you could get an easy cap for like 10€ but the drivers are ass to install but if you manage to get it working its fine.
mhei
i dont know if you can "get it back" but they should still show up in ytstudio on that video
go to livesplit settings (or was it layout settings) and make it always on top use obs studio to record. in it use window capture on livesplit and gamecapture on minecraft.
Mission: Escape from Island the run literally is hold W and spam Space to jump up hills. oh yeah you might have tho pres M1 to shoot couple times
i used to run the waveland demo in which after you complete the last level you would close the game and reopen because it puts you at the start of the hub world where you have to enter a thing to end the run. i see no problems with closing and reopening the game. skywardsword and tp(?) also have a glitch involving resetting the game (as you die)
i recommend to try twitchleecher some time