We don't get to pick our ancestors. Everyone must have ancestors who were born out of wedlock if you trace back enough generations. I have one a mere 12 generations back (I don't know his year of birth but he was more or less a contemporary of William Shakespeare).
Probably most Europeans are descended from Charlemagne, whose grandfather Charles Martel was born out of wedlock.
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.