Can you name a "well-rounded" assessment that doesn't just end up being a measure of privilege?

With these tests, you are still selecting for privilege, just differently.

More stable situation at home (so a better study environment) tutoring and, especially for language-based "soft" exams, gravitating in the right social circles will teach someone the "right" culture to reference in his essays...

At least for tests like the SAT, this is not correct. SES and coaching don't affect SAT scores much. A major reason why elites hate it so much and instead want "wholistic evaluations". Instead the ceiling of the test should be raised.