...selling the NHS etc...
This is another policy line the new Labour leader desperately needs to drop. It was just another one of the things that diminished Corbyn's credibility.
The real issue is about
funding healthcare. Corbyn tried to develop the issue to appeal to the anti-US/anti-Trump sentiment in his loyal following.
For all his waving around of documents in press conferences, the evidence was lacking. The best I could discern from it is a trade deal with the US may result in the price of some medicines going up. That is an issue about funding, not about "selling off the NHS".
US companies may well get (more) involved in providing NHS services, as a whole range of private companies already do. And Labour are not squeaky-clean on this issue thanks to the Blair/Brown PFI disaster(s).
A new Labour leader will need to acknowledge that private healthcare is not the exclusive preserve of the mega-rich. Labour voters use private facilities as well. There are small private hospitals/treatment centres popping up all over the place, and if the NHS has sent you to one of them for an operation because it is cheaper - and you've received prompt treatment in a clean modern building - then the bogeyman of "selling off the NHS" doesn't have the same bite.
And if Labour's appeal is going to be to London and comfortably off socialists rather than the working class, then the new Leader will need to have a think about how many of them already have private health insurance or some form of workplace healthcare benefit.
It all comes back to where Labour want to position themselves. As a far-left protest group that cannot get elected, or a centre ground party that develops policies based on practicalities rather than dogma.