The experience of doing a postgraduate degree is so far away from the experience of studying at Oxford as an undergraduate that they may as well be different universities. Different demographic makeup (much fewer private school kids, much more international students), different teaching methods (no training in 'blagging' a 1:1 tutorial with minimal reading, a very transferable skill to politics) and no time to hack around at the Union or make a name in one of the political clubs.
Honestly, the problem isn't even with all Oxford undergraduate degrees, just a slice of PPE and adjacent students who are vastly overrepresented in politics.
(My background: Leeds BA, Oxford MSc/DPhil. Don't like Starmer, though, despite our similar university choices...)