It wouldn't be a bad thing as such except that it (and the public schools like Eton that are disproportionally represented in Oxbridge) all to often represents patronage and nepotism rather than merit. (Oxbridge entry is NOT a meritocracy and neither is success in politics, government or a fair number of highly paid jobs).
And on education, I rated Joe Strummer (when he was alive) as having a better view of world politics than the vast majority of politicians.
Indeed. Not only that, by limiting the recruitment pool to only those institutions, you get politicians by default instead of the best candidates. Look at Diane Abbott - Cambridge educated apparently; yet is practically and clearly as thick as two short planks. Nobody can tell me she's the best Labour can find for the role she's in, yet there she is, because she hung around long enough and had 'Cambridge' written on her CV.