Wanna test yt-dl on their network
It’s possible. It depends on how sophisticated the IT department for your school or district is and how seriously they take security.
They ain’t know shit unless it’s automated
It’s technically possible but the chances of your school taking IT security that seriously are pretty much zero.
More likely there would be some sort of per pc bandwidth restriction in place.
If there isn’t and you max out their bandwidth somehow then they will notice and investigate.
Less prosperous schools may be managing with less bandwidth than some rich folks have in there home. The IT department wouldn’t be doing it just to be a pain they have to meet the needs of all of the users.