Yes, but whoever hits the button last gets control, so you can take priority back.

(Unless you manage to hit the priority button for 40 seconds - but I assume that's for locking out a broken stick that's doing inputs at rest)

Yes it will announce "priority right", but it's not normal to take control from the captain (and also highly unusual to take control from the first officer outside of training). Unless you're avoiding an immediate safety threat you would normally call out the problem but not take control.

It's a bit different for a training captain like in the video. They know these are the first officer's first landings on the real aircraft, so when things go wrong they take over.