Boris Johnson is so not any kind of political role model!
It doesn't matter what Gransnetters assume about whether Starmer stays or goes as PM. That decision rests with Labour MPs (unless Starmer decides himself to resign) and the Conservative Party. Eg If 80 or more Labour MPs wish for a vote of confidence in Starmer as PM, he will face a no confidence vote- which he may win or lose. The official opposition party, ie Conservative can also trigger a motion of no confidence, by way of expressing lack of confidence in the government if the May election result are dire for Labour. To do that only 15% of Conservative MPs (not a huge number currently!) write letters to the 1922 Committee. Badenoch is currently on the war path against Starmer so this is a possibility following the May elections.
If Starmer loses a no confidence vote, the government will then have a 14 day period to form a new government or face a General Election.
These scenarios may happen without any intervention whatsoever from Burnham if the May election results are bad for Labour.
Burnham, it is reported, plans to publish his "manifesto" plan for the Labour Party to turn around its current low fortunes directly after the May elections. If seen as "credible" that might mitigate the appetite and need for a no confidence vote. So one way to look at it, if credible, is that Burnham presents a potential solution. Another way to look at it is Burnham's plan is unforgivable, naked ambition and opportunism, so matter how good or bad or how credible his manifesto/plan is, it will be rejected out of hand regardless.
If that is the case, Starmer is on his own and will stand or fall at the hands of his Labour MPs.