It feels like the McKinsey-style design of large orgs has permeated the public sector just as much if not more than the private one. It always comes with a massive amount of procedure – employee surveys, follow-ups and meaningless 10-scale measurements, so that leadership can rest easy knowing that every decision they make is data driven. Somehow a spreadsheet analyst should be able to diagnose and fix every possible issue in an incremental manner, and conversely, all poor decisions are due to insufficient data.
E.g. "Do we think this form being filled out is worth someone not being able to see a doctor?"
Admin bloat seems to be at a higher level if it exists. Because ordinary secretaries are long gone.
The obvious solution seems to be reducing the required paperwork rather than hiring more staff to handle the paperwork, and this seems to agree with your experience when you described "unintegrated time recording systems".
That said, I wouldn't be surprised if too much bureaucracy was also partly to blame, because that's what happened here in Canada.
Because governments have constantly demanded the NHS spend less money, but have also been under pressure to preserve numbers of "front line" staff you've ended up with serious shortages of low level admins, medical secretaries, receptionists, finance officers etc. This means that more and more healthcare workers spend their time doing paperwork or customer service work that could be done by anyone. I worked out of one of the largest genomic diagnostic labs in the country with over 150 clinical geneticists, nurses, scientists and consultants and there were 4 admins. That meant nurses were staffing reception, consultants were managing their own diaries, geneticists were doing stock takes and clinical scientists were packaging up and booking samples onto DHL for collection. This was not an efficient way of working, but the lab had a hard ban on hiring any non "front line" staff. And it had been running like that for years.
The trust as a whole had the same problem on a macro scale. There weren't enough finance officers so invoices didn't get paid or debtors chased (the NHS provides a lot of private services as well). Because of a lack of staff working in recruitment it could take months (4 in my case) between being given a job offer and the paperwork being in place to actually start. Payroll was a nightmare because of lack of staff and multiple unintegrated time recording systems, many of them to this day paper based.
A large number of legacy patient records are still paper based and have to be physically moved around and digitised "on the fly" but because the NHS is supposed to be digital there aren't enough porters payed to literally push the trolleys of records around so you end up with nurses doing it.
I could go on for a long time but you get the idea.