This hasn't aged well, has it. Starmer is clearly hanging on.
I think you are confused about the rules. The rules do not say he has to finish his mayoral term before he can return to parliament. They say, " Directly Elected Mayors and Police and Crime Commissioners must seek the express permission of the NEC/SEC/WEC (as applicable) before seeking nomination as Labour candidates for the Westminster Parliament" - Rule 5.IV.2. Burnham therefore needed NEC approval to run in the Gorton and Denton by-election. Starmer and his allies on the NEC blocked him. That isn't media spin. That is fact.
If Burnham is prevented from returning to Parliament until his term as mayor is over, it is unlikely he will become Labour leader prior to the next general election. The election must be held no later than July 2029, although it is likely to be earlier than that if Labour's poll rating picks up. Burnham's term as Mayor ends in May 2028. It is unlikely Labour would want to change leader with only a year to go to the election.
In terms of where we are, it is clear that, even if Starmer manages to carry on, he is fatally damaged due to the number of MPs who have openly called for him to go. The Labour party is historically bad at regicide so he may be able to struggle on, but at the moment the contest seems to be between those who want a "swift" timetable for his departure (Streeting's supporters who want a quick contest whilst Burnham is not eligible) and those who want an "orderly" transition (a slower change of leader that allows Burnham to return to parliament).
Streeting and his supporters clearly want a quick contest as he is unlikely to win if Burnham is a candidate. However, the left and soft left want Burnham, so if there is an early contest they may put up a "stop Streeting" candidate. I understand some are suggesting that Ed Milliband could be that candidate.