PHE is an executive agency of the Department of Health and Social Care not an autonomous quango as you state i.e it is part of the Civil Service and staffed by civil servants and Matt Hancock is directly responsible for it, its responsibilities, budgets and plans. Though to be fair if you got your information on its governance from, say the Daily Mail then you could be forgiven for thinking it is a quango sitting far more outside the control of Ministers.
Fortunately there are more informative sources such as the Institute for Government which explain its relationship, how it was established by the Cameron government and how its integration within and accountable to a government department abolished about 70 of those nasty autonomous quangos you reference.
https://www.instituteforgovernment.o...health-england
That seems less pertinent than who is responsible for the testing and tracing programme though. As previously discussed this is led by Dido Harding. She reports directly to Boris Johnson and the Cabinet Secretary (if we had one) and not to the Secretary of State for Health and Social Care. (see Government Announcement). She is supported in this role by John Newton of PHE.
https://www.gov.uk/government/news/n...amme-appointed
Other than Newton's supportive role it is not clear where PHE fits in the testing and tracing governance. What is clear is that it is not responsible for it. That sits fully in the hands of a Tory Peer reporting directly to the Prime Minister.
If you want to change your view that PHE is responsible for the 25,000 tracers who have contacted an average of less than two people each since test, track and trace was introduced from a simple yes to a simple no now would be a good time I guess.