The Government is responsible for funding and the structure of the NHS and for it's performance.
If the NHS has failings in these things then it is the Governments job to sort it out.
PHE and the NHS do the bidding of the Government. They do not decide on how much brass they receive or how the service is deployed. That's up to the Government.
They advise the Government but it is the Government who make these decisions.
That is why there is so much chat at election times around the NHS. "24 hours to save the NHS"
At election time you must decide which political party has the best plan for the NHS, and everything else.
The Government does not deserve a free pass if things that are under their control go wrong. Is the budget sufficient? Is the structure right? Their performance needs scrutinising.
We don't vote for members of PHE or NHS management staffing, we vote for a Government who decide these things.
That's how democracy works in this country.