Which of May, Johnson, Truss would you have wanted serving their full term?
Burnham does have a mandate, as the mandate was given to his party and he's the leader of that party. That's the system we have, and anyone ignorant of it is either new to politics, or willfully ignorant. We have representational party politics, not a presidency (though of course, the PM carries a lot of weight, and sets tone for the party and government)
We may not like it, but we can't pretend that we don't have that system.
I've long held that an incoming new PM OUGHT to call a GE within, say, 6 months, but I've never believed that it was actually compulsory, and I don't think it's ever happened as a point of principle (as opposed to cynical expedience).
TBH, the leadership of Johnson, and especially of Truss has made me much more wary of making it more difficult for a governing party to take out their own leader. Though I do think that all (major) parties - say any party with 10+ seats should have the same mechanism for removing (and selecting) their leader.