I think you need to wind back a little.
We had a Remain PM who said he would stay on, invoke article 50 immediately and see the process through if the UK voted to leave.
He bottled it and ran off.
We then had a Remain majority Tory set of MPs who gradually whittled away (with help from back-stabbing Gove) at the Leaver candidates until we were left with May or Leadsom.
I like Leadsom. A thorough nice lady, but not in my opinion a satisfactory leadership candidate, like May.
Leadsom received the vote of 84 MPs, less than 25%.
She was then lent on not to take it to the membership.
Had Davis entered the contest he might have done better, but I still think the MPs would have gone for May in such a number that the person coming 2nd would have felt the need to step aside.