To my mind, there is something very disturbing about a process that means the Prime Minister is elected by a tiny minority of voters, who happen to generally have pretty extreme right wing views that are completely unrepresentative of the electorate as a whole - meaning that the two candidates have a strong incentive to make promises that pander to those views. And then the winner almost automatically becomes Prime Minister of the entire country, without anyone else having a say in the matter. The system was in principle just as bad when Gordon Brown became PM
But, as you acknowledge, that's how it works. It's for the Government to decide who will be Prime Minister, and how they want to make the decision.
This is the 4th time in (my, at least) living memory a Prime Minister has resigned during a Government.
Thatcher: replacement chosen by Tory MPs
Blair: replacement was pre-chosen
Cameron: replacement chosen by Tory MPs
May: MPs followed by membership ballot, likely to be a rubber-stamping exercise
Before that you need to go back to Chamberlain I think.
I realise to some extent this is the inevitable result of our system in which the Prime Minister is not directly elected, and would be quite hard to change while keeping the essentials of our Parliamentary system, but the current contest does seem to show up a big problem.
What would an alternative look like though? An automatic general election? Feels likely to mean unpopular PMs have to cling on even if their own party no longer wants them.
A presidential system? That'd be a massive upheaval, and in many presidential systems you'd have even less involvement - the deputy would just take over. Who would May's deputy president have been? Hmm!