I’ve got at least one of those in my attic, but the father had not married (uh, I think) and was responsible enough that he made sure his blessed accident (actually at least two) were taken care of. Elizabethan era, like yours. And it’s merely family folklore, I’ve never done the legwork.
Given what happened to some legit birth folks who wound up doing pretty well during the Tudors, it may have been better to be born out of wedlock.
My illegitimate ancestor was the son of a nobleman of a small city in Europe (not in Britain)--the nobles seem to have had lots of illegitimate offspring but often took care of them. In the case of my ancestors they were set up as architects.
European royalty tended to have lots of illegitimate children by their mistresses--during the reign of Charles II of England, people who didn't want his younger brother to become king made an unsuccessful attempt to have one of Charles' illegitimate sons made the successor.