You are being - and I do not mean this in a bad way - pedantic in the wording of accountabilities Vs responsibility for action/provisionIt is clear that overriding responsibility lies with the Department of Health and Care and specifically the Secretary of State (unless I have misunderstood the documents you posted).
Contracts that I have put in place for government departments may have required an appropriate SoS to sign them as they are the accountable officer - but the work of producing them and managing them will have been with the Home Office, HMRC, DWP etc.
So in this case you will see the PHE's framework include statements like:
"Fulfil the Secretary of State’s duty to protect the public’s health from infectious diseases and other public health hazards, working with the NHS...…"
That is an accountability of the state - embodied in the SoS - which is discharged through DHSC and onwards to NHS or in this case to PHE. The responsibility firmly sits with PHE.
The SoS is also 'accountable' for the commissioning of every care home - but the responsibility for doing it is done by a chain of responsibilities - DHSC - NHS - CCGs/LAs
The The SoS is also 'accountable' for the procurement of every mask and other item of PPE but the responsibility for doing it is done by a chain of responsibilities - DHSC - NHS - NHS Procurement and NHS Supply Chain
Technically, the SoS is responsible for every poor quality sandwich served and bedpan not removed - but we would not expect that SoS to be spending all day and every day on those matters.
Responsibilities for planning and delivery of services are discharged through Departments or other bodies.
As mentioned by someone earlier 'technically' the PM and an SoS can be blamed for every public service issue - but there needs to be a level of pragmatism and realism applied to how these things are actually managed/discharged