I agree, and just to add there are differences at a statutory level.
In particular, s 1(1)(f) of the Legal Services Act 2007 has the objective of "encouraging an independent, strong, diverse and effective legal profession".
The SRA was created under the LSA, so it's absolutely woven into the SRA's DNA, and hence the profession as a whole. In addition, DEI policies can help firms avoid discrimination claims under UK law.
I'd add, incidentally, that I don't think graduate recruiters necessarily are very committed to any apparent beliefs: for example, a "well-being ambassador" and HR director at CMS was fired this week after posting "I hate Islam" and similar comments on Twitter. I'm sure he expressed his commitment to diversity in his day job, whereas in reality he (not-so) secretly despised it!
Fundamentally law firms want to make as much money as possible, and the extent to which pro bono, DEI, etc. are there to meet regulatory requirements/improve their brand image, as distinct from something they genuinely believe in, is difficult to prove with any certainty. However, for every Dan Neidle, who has retired from his 7-digit partner role to do something he believes in, there are 500 more partners who keep going 'til retirement, so my money is always on "it's all about the money". And with that said, there doesn't seem to be any money in conspicuously disavowing DEI as a law firm in the UK, and plenty of money in continuing to profess your commitment to it.