I don't personally like the public/private debate in relation to anything.
In healthcare the comparison is always (and perhaps understandably) made with America. During Covid pandemic, the performance of our health & social care system was compared unfavourably with particularly the German, Japanese and South Korean systems. But of course none of those three countries operate a free-at-the point of service universal system. They do all supplement their essentially private health & social care systems with state-backed safety net for those that need it and all three offer consistently better health & social care outcomes than we receive in the UK.
I don't have an aversion to public ownership, of for example, utilities or rail, but as a child of the 80s with Scottish mother and London the constant trips to see family using British rail and memories of constant power cuts growing up tell me that public ownership isn't a panacea to fix all problems.
The important thing for me isn't ownership or sector, it's how do we deliver good outcomes.
On public sector pay, we need to talk about the elephant in the room: pensions. Public sector pensions are defined benefit schemes underwritten by the government. So if you pay nurses £73K a year you are committing to underwriting what will likely be hugely significant scheme shortfalls for what will end up being millions of people (when we say there are 800K nurses in the UK obviously this is a revolving conveyor belt of people moving in and out of these posts and picking up pension entitlements as they go).
So the cost of paying a nurse £73K a year to the NHS is an order of magnitude higher than the cost of a private sector employer paying someone the same wage.
I don't believe the alternatives you've put forward are a different system they are all possible in the current "system" and as per reference to BR, public owned utilities etc, have all been in place in previous decades.