First, a meta-point about questions like “why doesn’t the US have policy X Y or Z”: The US is essentially set up to be ungovernable by design (any meaningful reform requires the cooperation of both major political parties, a situation that doesn’t exist anywhere else in the developed world), you can’t really find rational reasons for most policies.

I notice this a lot, actually: the fundamental thing Europeans miss about the US is they think “a rule exists, thus it must be there on purpose”, because in your countries, unlike in the US, you have functioning parliaments and governments that can change laws as necessary.

Anyway, the US arrived at its education system, as with most things, through a long and largely random process. The catalyst for the weird non-academic admissions standards in particular is well-documented to have been pure anti-Semitism: college administrators felt that with purely academic admissions standards, too many Jews were being admitted.

So a field specific exam doesn’t really work.

And, besides that, would be opposed by the exact same people opposing the SAT.