The one you refer to isn't really 'private'. It's part of the NHS, with the doctors having similar contracts to other GPs. (After all, no GPs are actually employed by the NHS. They're all self-employed and have simply entered into a contractual arrangement to provide healthcare services to NHS patients, just as the doctors in your link have).
well.....while I agree it has its limitations (and the service won't take on all patients) my own GO surgery already uses some of the same ideas and it works very well.
Would you get continuity of care - ie same doctor each time? It's hard enough at the surgery I attend now, but at least I can mostly specify which one I want to see and avoid the one I don't.
If it doesn't work, how easy will it be to find a regular surgery to take you on? Given that there would probably quite a lot of homeless patients all at once.
If I want a doctor it's likely to be because I want to see them not to call them for a chat. It's too impersonal; human "contact" isn't the same over tech. (My last employer totally failed to understand this; I strongly suspect deliberately.)