The overall context is a government that has been openly antagonistic to the NHS for at least a decade, pushed for disastrously expensive "privatisation by stealth" efforts [0][1][2], and is now effectively trying to outsource the digital side of NHS operations to private companies [3]. Strategically starving certain departments is par for the course.
[0] https://www.theguardian.com/society/2014/apr/04/gp-local-nhs...
[1] https://the-probe.co.uk/blog/2019/12/hancock-urged-to-halt-n...
[2] https://www.bmj.com/content/346/bmj.f1322
[3] https://en.wikipedia.org/wiki/Babylon_Health#Matthew_Hancock
(and many, many more sources if you really need them...)
If the goal is to "starve the beast" NHSX isn't having trouble with retention then are they?
They are a very large provider, and that's causing problems for their hosting CCG.
None of this detracts from your point: the NHS has been deliberately under-funded for years, and the Conservatives are pushing for non-NHS providers to provide services.
And yet I didn't see any obvious job openings. I didn't see an obvious way of checking civilservicejobs.service.gov.uk as neither NHSX nor National Health… turned up any results when plugging them into the organization filter. It's an interesting mission, but if they're not hiring then I can only imagine they're starved for funding or they've successfully retained enough folks.