NHS England also run centralised IT services (e.g. the NHS spine which contains patient data), and runs pretty much all the education of all staff, including clinical staff across the NHS.
They also have responsibilities relating to research and actually putting together information from all the local hospitals so e.g. Wes Streeting's team can get information in a format where they can actually analyse it.
They organise the delivery of national immunisation programmes and making sure e.g. COVID vaccines are where they need to be, when they need to be.
On top of that, they, working with ICBs which are also experiencing massive cuts, work out what services need to be offered to local populations (e.g. where is big enough to need an A and E, should doctors surgeries be allowed to close or do they need to be run by someone else, where should a minor injuries unit be, where should eating disorder patients be treated) and work out how to make sure the population has access to specialised commissioning such as proton beam therapy and how to manage tertiary (super specialist) hospitals.
They also work out how much money different areas and different organisations within different areas should get - boring but someone has to do it. They make sure local pharmacies have the contracts to do our prescriptions and that we have enough pharmacies in our local areas to meet our needs.
They agree and organise care for people who need complex, long term health care but who don't need to be in hospital.
You can't procure healthcare for 67 million people without the things above being managed. And that involves back office staff.
The organisations above have already been asked to make big savings under the last government (around 20-30% by the end of 2023-24).
Another 50% will mean some services have to be stopped.
Editing to add: they also procure all the medicines used in the UK's hospitals. I'm sure I'll remember more.