Correct. One of the defining features of the Aspire contract that is - agonisingly slowly - being replaced is that it was a procurement framework as much as it was a contract for delivery. The real genius - from Capgemini's perspective - is that Aspire personnel (i.e. mostly Capgemini staff) ran the processes of contracting and procurement from the Aspire suppliers (i.e. mostly Capgemini and Fujitsu).
This wasn't abused anywhere near as much you might immediately assume because that element of the contract was well-monitored by HMRC and it was run on an effectively zero margin basis. But in 2014 when this contractual structure was broken up (the "novation"), many of these key procurement/contracts personnel were offered a reverse-TUPE back into the civil service, and the overwhelming majority declined and moved onto other roles.
(For most the concern was they'd be stuck in one role forever; TUPE would of course maintain their private sector conditions, but if they ever wanted promotion/transfer they'd almost certainly have to adopt civil service Ts&Cs)
The other element of this is that HMRC are sitting on the business end of ten years of austerity, plus being one of the most badly affected departments by Brexit, plus having to move heaven and earth to implement Covid response plans like the CJRS. Even if there were budget or the people to do so, there simply isn't the organisational appetite to implement significant changes to suppliers.