If I remember correctly, the public sector pay cap was put in place after the 2010 general election because it was seen as unfair that public sector pay rises were continuing much as they had before the 2008 crash, whilst many in the private sector had taken quite substantial pay cuts in order to keep businesses afloat. If private sector pay has regained the losses it sustained after 2008 and is now increasing faster than public sector pay, it would indeed be fair and right to see rises in the public sector. However, I'm not sure that's the case, so the case for public sector rises may not be as strong as some might suggest. Some measure of fairness does need to be maintained between public and private sectors.
I'm not too sure what the actual comparative figures are, but it's certainly something government should look at. Just giving in to whoever shouts loudest is no way to be fair to the majority.
The NHS needs reform. It's not good enough. That's no reflection on those at the sharp end, the vast majority of whom do their best every day (and their best is often very good indeed), but they sometimes do their best despite the system, not because of it. However, the system has absolutely no incentive to do better, because there's no competition to take work away from it. Until there is, we'll have to carry on putting up with a creaking bureaucratic machine lurching from crisis to crisis. I don't see things improving any time soon because there's too much else on the political slate at the moment, not least the ramifications of delivering Brexit. Maybe when that's settled, we could have a proper national debate about what we expect of the nation's health care provision, and what can be provided, how, and at what cost.
The current system has been in place for 70 years, now. It used to be the best in the world, but that's no longer the case. Maybe it's time to start thinking about how we could do things better. (I do, however, stress again - most strongly - that's no criticism of those at the sharp end of heathcare provision. It's about the system, not the people working in it.)