There may be some parliamentary country where what you say is correct, but in general, yeah, no, that’s not how it works.

Remember Liz Truss, all 49 days of her? A PM who fucks up on a Truss/Trump scale generally finds themselves very rapidly seeking alternative employment. Truss was forced to appoint a borderline sane chancellor about two weeks after causing the bond yield to go crazy, and was gone within another couple of weeks.