From my experience of interviewing, once you get to KCL (5th downward) your employability will be the same... Nobody is going to discriminate against Southampton V Liverpool.
What I found out in work is most people are genuinely impressed if you did STEM at Oxbridge (a small 10% of usually the far-left activists type will be hostile to you because you are part of the 'privileged establishment'), their facial expressions don't lie. I've tried to say I've an MSc from UCL a few times, and the reactions are significantly bland.
I've been on this forum for a decade or so, every year there are sour grapes on TSR who say university doesn't matter, from my observation those tend to be high-frequency posters or forum admins still studying at some third-rate uni. I believed this BS when I was naive: You can still see my comment from 2016 that said "If I had a second chance, I'm would not go Cambridge."
But when I got to the job market, I cannot tell you how much this has helped my career. Cambridge has been my golden ticket for life, especially applying to junior roles where competition in the job market is so fierce today. I know for a fact my first two jobs I wouldn't had the offer had I not had Cambridge on my CV. Yes it's not the be-all-end-all, and how far you get in life is still up to you, but it's a pretty darn good 30m head start in a 100m race. I do not envy my friend from UCL who did 300 job applications, I did like 30 and did not put that much effort into CV and techniques.
In general Oxbridge will give you a halo, LSE for the financial sector (though not as prominent today) and Imperial for academia research. Sure it won't matter as much 3-5 years down the line, but being Oxbridge will help you get on a big org for your first job and generally get promoted faster. The only true place that doesn't judge your degree is the civil service, and you better apply to a role that doesn't require a CV, then good luck standing out amongst 250 applicants for the same job.
I expect certain individuals to rebut my opinion, but 1.They never been to Oxbridge and 2.They are just saying this to make themselves feel better. Believe whatever you want to believe, I've told you my life story and this is the story repeated generations after generations with people paying £100k a year for private schools just to give their children an edge into Oxbridge.