@Funcone I think it's fair to scrutinise any government data, ONS or otherwise. That said, I do typically land in the cock up camp, rather than conspiracy camp when it comes to the figures. Often, the damage comes from wilful misrepresentation of the figures, or an assumption to fill a void left by a lack of data.
The civil service has always been too departmental in its approach. For example, for decades DWP paid benefits to the population without a real understanding of what claimants were earning because DWP & HMRC operated in a vacuum. Checks were done retrospectively, if at all. Now, with the further roll out of Universal Credit (with all its problems), those two departments now share data in realtime because it's necessary for the payment. Change like that can happen, but often only as a by-product policy rather than a determination to improve.
You can usually trace data issues back to the source, with the LFS that you referenced, it is imperfect, but it's pragmatic. The alternative would be a hugely expensive and intrusive data collection from every business in the land. I suspect that there isn't the political will to do that, not least because it'd cost a lot, but also the work involved in doing it and chasing up non-compliance, it would be a minefield.
You might also argue quite fairly that better data on some fronts is not to the advantage of a government, especially if is to be compared to decades of older data that was collected by less accurate means.
In the NHS specifically, Foundation Trusts were brought in to provide greater autonomy, to allow decisions to be made more locally including procurement. Therefore, you had a situation where Trusts were buying all sorts of IT systems that didn't speak to each other that all stored data in different ways, making it extremely hard to provide a national picture. In part, the solution to this was to be the
Lorenzo system, which is a textbook example of a disastrous IT deployment.
A lasting win from a the pandemic will be an improvement in NHS IT. Not only things like a functioning NHS app (again, got to where it is as a means to a Covid pass) but an understanding that this sort of stuff can be done which will hopefully spur further innovation in the future.