That model absolutely works with car mechanics. I've got a mechanic that I go to for 80% of my needs. But I get tires from someone else, as well as oil changes. I do some stuff myself. When my maintenance interval service comes up, he does it, gives everything the once over and does anything he's found that makes sense for him to do. If it's transmission related he sends me to the transmission specialist.
I've had a 20+ year relationship with my mechanic.
Not only this...your mechanic/doctor will be in a competitive marketplace just as much as all of the other mechanics/doctors. It's not like he's going to be the lone holdout.
Of course you'd still be able to keep the Docs you wanted to keep. The idea is making healthcare more competitive. That doesn't necessarily mean you have to bounce around. How much do we bounce around with things we buy in competitive marketplaces? Do you travel across town to the other supermarket every other week? Or do you unwittingly benefit from the fact that your supermarket has to have competitive prices?
Moreover, the biggest problem isn't primary care. It's with specialty care.