why won't they just kill the king and bishop?
Because that would just cause more conflicts.
Most average citizens would not be aware of politics, so if the king was suddenly killed, it'd be easy for power-hungry nobles to start small rebellions all over so they can try for the empty throne.
It'd be better to have a public trial, to show the people why he is being dethrones, and then formally crown a new king at the same time. If the people are made aware and shown that the takeover is reasonable, very few would complain.
Also, by being non-violent, any opposing faction that tries violence in the power-struggle would look like a villain to citizens and get very little support.