I think I need a video of you making aeropress! It's super clean for me. Some suggestions:
> I always manage to spill grounds getting them into the Aeropress,
Use the scoop but don't pour or tap them in - instead put the scoop into the cylinder and then spin the handle to dump out the grounds. Makes it impossible to spill.
> Plus there's the risk of your cup flying away with the amount of pressure you're putting on the top of the press
That sounds like way too much pressure - some new aeropresses can be tight (which can be remedied) but maybe you have too fine a grind. It should definitely not be espresso level which requires that type of muscle. It's not brewing by the pressure you are applying but time + agitation. The press is just removing the grounds.
I have however frequently knocked over a pourover while pouring... my own fault... tend to use a heavy ass kettle which turns it into some weird strength endurance exercise.