People bayed for Starmer's blood from the start of his premiership - both the media and the public who believed the partisan reporting and the incessant slogans. They got their way, and Starmer stepped down. Someone has to take over - what did they expect would happen?
Burnham returned to parliament, presumably a he and others knew that he stands a chance of reuniting the party and keeping the country on the road to the fairness and equality that was laid out in the manifesto. Was his ambition any more or less reprehensible than that of any other MP or Cabinet member? I'd be interested to see the reasoning behind that if anyone who thinks so would like to share it.
Others could stand against him if they wished, but so far nobody has come forward to do so, although there is time for that to happen.
If someone does come forward there will be a contest, the result of which will be decided by the LP membership, as is usual in these matters, and as happened in the Conservative party when they deposed and re-elected numerous leaders in their last stretch of government.
If not, and Burnham is the only candidate then he will win, and become PM. He will continue as MP for Makerfield, as the people of Makerfield voted him in, above the 13 other candidates who stood against him. He, like all of the other MPs in the UK, was democratically elected to the role with a resounding victory.
The rest of us will keep the MPs we had before, with business as usual - some of us voted for the ones we've got, others would have preferred one of the other lot, but that's democracy.
Johnson, May, Truss and Sunak all took office without general elections, if I remember rightly? What's the difference? To answer the OP, the democratic process is that the person elected by Party members leads the Labour Party, which was elected into government in the general election of 2024, just as happened with a succession of leaders of the Conservatives.