Yeah, but no one else gives a shit who runs the local tennis club, or has a stake in it.
Personally, as an LP member, I do not believe that a) I should have more say over who is PM than my neighbour who isn't, and b) that I'm as qualified as my MP to decide who can both lead, and build a team that will deliver (broadly) a manifesto, given that that they know all the candidates, and I don't.
The job of the PM is to both lead, and serve, Parliament, with MP's as the representatives of their constituencies and electorates - not, and I include myself in this - the 60-odd weirdos out of 80,000 voters who turn up to Friday night constituency meetings.
For me, for Parliamentary democracy to work, MP's should be affiliated to a party - or not - but not owned by them, or controlled by them. They are there to represent all their constituents, not just the ones who voted for them, and certainly not just the miniscule number who happen to be members of the same political party.
By all means look at divorcing the roles of party leader and PM, with the party leader role being very much the one controlled by the membership - but the PM's role is very different, responsible to a far wider electorate, requiring a different skill set and having infinitely more serious responsibilities.