No, the public sector could never have delivered the T&T system alone.2 separate issues being rolled into one.
With regards to pay rises, I would say that no one should get any rise until all this is over. We can then see what the economy looks like and spend what we have in the best way possible. The furore over nurses is just the Trojan horse for pay a wider discussion on pay that has far wider implications.
With regard to Test, Track and Trace at some point the breakdown of what's been spent where will come under a fine toothcomb, but make no mistake an industry that can manufacture, deliver and utilise 1.5 million tests in a day and then gnome sequence the results is never going to come cheap. Much is being made of paying consultancy staff £1000 per day, but the in the throes of a pandemic do you realistically think public servants could have delivered this on their own? The costs also pale into insignificance when compared to the 100's of billions of £ thrown at all aspects of the rest of the pandemic.
With regards to heads rolling, I suspect that in normal times ministers would have left office, but in these troubled times I think there has been a concerted effort to hold the line. If one minister had fallen, many others would have gone too and it would have been a revolving door. Getting to the end of this pandemic is more important than seeing heads roll in the govt's view and you will get to have your say at the next election
But one of the massive mistakes at the very start of this pandemic - among many - was that Boris and his chums automatically turned to the ‘usual suspects’ to set up and run the system rather than utilising the existing massive expertise and knowledge that LAs and other parts of the health service possessed.