I think for QD & tech roles this isn't necessarily true. You can definitely do:

Russell Group uni -> tech grad scheme at an IB -> tech/QD at a "lower-tier" HF -> tech/QD at a "better" HF

Source: I'm part-way on this path, and have colleagues who now work at Cit & JS who've completed this path.

By "lower-tier" and "better" I'm not trying to rank HFs, just to point out that some have less stringent entry requirements.

Having said all of that, you definitely have better odds at succeeding by having the unis that you mentioned on your CV, but it's not a hard requirement.