Believe Cummings point was that The Department of Health wasn’t in control as it is in terms of test and trace. So NHS + Cabinet office (and not DHSC’s normal red tape, funding models) = success.
However, I think “sources” in the vaccination task force have said Cummings is full of it.
“That is why we had to take the vaccine process out of the Department of Health.”
Q193, https://committees.parliament.uk/oralevidence/1911/html/
I therefore don't in any way trust his self-assessment of his own department's work and the 'vaccine task force' he set up, which incidentally hired some of his associates, who were paid very well, for example the PR firm:
https://www.ft.com/content/6159d5b7-171c-49ed-b318-144efcc4c...
A director of the public relations firm paid £670,000 to advise the head of the UK government's coronavirus vaccine task force is a longstanding business associate of Dominic Cummings.
According to Dominic Cummings recent select committee hearing, the vaccine programme was largely built outside/adjacent to any existing NHS processes. It was given NHS branding for public trust reasons (just like the new-from-the-ground-up Track and Trace endeavour).