If I was being pedantic I would have pointed out that the quote you mentioned appeared in a Guardian article about Ministers seeking to transfer blame and responsibility to PHE. Rather than the selective quote you seem to give it was the Prime Minister's spokesman who made the quote.
https://www.theguardian.com/world/20...ovid-19-errors
Asked whether PHE was under threat, the prime minister’s spokesman praised its role in the coronavirus crisis, but also stressed the need to ensure that institutions were “fit to cope” with future emergencies.
“Public Health England are playing a key role in our response, working on important issues such as detection, surveillance, contact-tracing and testing,” he said. “They have been working hard to help protect the country and to provide insight in our efforts to beat the virus.”
The spokesman added: “We’ve made changes to structures already, for instance we have set up, from scratch, NHS test and trace, and the joint biosecurity centre. Any future changes that we might make would be aimed at further strengthening our public health capability.”
When seen in that context it is obvious that the Prime Minister's spokesman when using the term we is referencing Boris Johnson's government and not PHE. This perfectly fits with the governance in which Dido Harding set up the test and trace system from scratch reporting directly to the Prime Minister.
I am sure you did not intentionally try to mislead to make your point though.