I think that might be too pessimistic. Firstly, the NEC needs to approve him. I think they might prefer a hammering by disgruntled Burnham supporters over the challenge to Starmer, we've seen they are quite happy to be anti-democratic and with the resulting mayoral election costing over £2 million, they could use the expense as an excuse.
If Burnham does run then I don't think there's a guarantee that he'll win. I think he stands a reasonable chance - he does have a good favouribility rating but the MRP polls demonstrate that Reform was gathering momentum (30) to Labour (28) in this constituency - and I think Galloway in particular has the potential to split that vote.
Then he won't be able to start a challenge until after the elections in May. The Starmerites will dig in and delay after that. They will engage in full blood campaign of character assignations leaving the party even more beleaguered. Probably over the summer holidays. Maybe a party election in the autumn? I bet Reeves gets her budget in early in this time line.
If Burnham wins out - then we'll have another round of the government talking about radical change only to rub up against the bond markets, the general public, the far more sensible, and crucially slow, Lords and mounting international pressure to prioritise defense spending. If it doesn't go well there will be mounting pressure for a general election.