Although legally fine, there certainly is some lack of legitimacy when PMs are elected by party members after the predecessor is axed, because the voters never had a chance to assess the person they end up with (in this role).

The larger issue is that the new Tories' procedure for producing candidates leads to a winner that has a large majority of parliament against her.

There is now a problematic mixture of parliamentary and Tory voter legitimization. That would be resolved by a true general election.