Big businesses don't really care, they care about what firm you are in, but the firms self-select based on achievement and alma mater. If you got oxbridge Undergrad or BCL/LLM, more likely the better firms will choose you, and therefore more likely you are exposed to bigger cases.

It's a self-selecting and circular thing. Between two people with GPA that are the same and extracurricular same, might as well choose the guy with the better school.

There are of course cases of delisted uni or SG uni in MC or white shoe (in fact I would say quite a few), but it's harder to make the jump.

Ofc, if you good enough to jump to MC or white shoe without oxbridge, then it doesn't matter, but it's easier to go MC or white shoe with that stupid BCL/LLM than without.