How valid are ghusl and wudu' according to the schools of thought that consider rinsing the mouth obligatory, if a tooth is broken and causes pain during eating, and one wishes to fill it or replace part of it?

There is no objection to installing or filling artificial teeth when needed. This is supported by the Companions' use of prosthetic limbs, such as the nose of Arfajah ibn As'ad. This does not affect the validity of Wudu' or Ghusl, and there is no obligation to remove installed teeth during them, just as there is no obligation to remove a ring during Wudu', given its similarity to it. The Prophet used to wear a ring, and it was never reported that he moved it during Wudu'.