I know one can ask Cad as duit/tu to ask where a person is from, but is it correct (politically and/or grammatically) to ask specifically where someone was born?
As I understand it, one cannot simply say "Ca rugadh thu?"

You mean, like you cannot simply say "What is your net worth?" :-)

Grammatically, there is nothing wrong with "Cén áit ar rugadh thú?" or "Cár rugadh thú?" as far as I know. But adding "ó cheart" (originally) to "Cad as duit?" is a less abrupt way of asking, I suppose.

Cad a deirtear sa nGaeltacht?

"Cad as duit ó dhúchas?" would be another way.