It’s permissible by the letter of the rules, but it absolutely is not what people expect to be happening with their democratic say.
Most people vote for whichever MP is in the party they want to vote for, run by the leader they want to run the country. Most people expect some stability, at least until the next opportunity to have their say.
And Labour party members are supposed to vote on the party leader, it isn’t supposed to just be a decision by MP’s. Because Burnham stood uncontested, there was no vote, and effectively the only people who got to decide were the MP’s at the nomination stage.
And Starmer had no good (publicly known) reason to leave. His controversies were never that damning, and they were contrasted with quiet competent successes.
You dismiss this as “all right and proper” according to British election rules, while ignoring that it is all dripping in slime.