Would you let your college age child study at a university abroad?

Yes.

If they're college age, they're no longer a child

My daughter went on a study aboard program for the summer. She took a couple classes at a university and lived in the dorms. Several students from her university went. As did a couple of her professors who were guest teachers for the summer at the university they went to. They were there about three months.

To attend a university in a foreign country for the entire length of thei undergrad studies, no I would not. Money would be one issue I would have with it. Safely another. Being so far away from them in any emergency or serious illness. And cost and time making seeing them way too infrequent. I am in the US so unless it was Canada, going to another country would be very far away.

IF they have the interest, the language skills for that country, have been accepted by a good school, and have some maturity to take care of themselves, why not? College is for spreading your wings & learning to fly. However, some young adults need baby-steps, like living at home & going to local university or community college. Depends how well you raised your kid!

Mine did and learned Hungarian which is a very challenging language.