Did they have debutantes in the elizabethan era?
I don’t think so. The custom of young women ‘coming out’ formally into society seems to have started, as far as I can make out, in the 18th century. By the early 19th century it was certainly an established custom, there are discussions about whether Fanny Price is ‘out’ or not in ‘Mansfield Park’ by Jane Austen for example. In Elizabethan times, young girls of the upper classes might be married off very young, twelve was the minimum age for marriage for girls, and fourteen for boys (of course many people got married much older than that, but it was possible to be married very young). So the idea of a young person waiting until they were grown up before being introduced into society probably hadn’t really caught on.