I mean I just feel like it's a pretty logical conclusion to say that unis which are most represented in big law firms are also going to be the unis which those firms target. You say prestige doesn't influence how represented a uni will be at law firms, I simply disagree. Of course no law firm is going to come out and say this, mainly because it would stir up accusations of "elitism". So since no law firm is going to come right out and say this, you can't ever "prove" it, but it's clearly the case. There are 'non-prestigious' universities with much higher student populations than some of those universities on the list (for example NTU has more students than Uni of Bristol) and yet they seem completely unrepresented in law firm graduate statistics. I feel like this disproves your argument of "the prestigious unis show up more often because law firms get more applicants from them", because if that's the case then where are all the law graduates from non-represented unis applying to? The fact that they can have higher student populations and yet not even show up on graduate statistics shows a clear link and I simply feel it leads to a logical conclusion that uni prestige does in fact matter and target unis do in fact exist.
For the record, I know there's more factors which influence your chance of landing a good job at a top law firm than just your uni, such as your experience (and I plan to get summer internships where I can for this sort of stuff), but I'm just making the point that there are very clearly unis which law firms prefer to recruit graduates from. If you really wanna tell me that you think someone graduating from Oxford Uni doesn't give them an edge over someone applying from let's say London Met, then we have nothing left to discuss because we won't see eye to eye.