In the real world, people are often invited into groups after they have shown some innate aptitude.
You mean, like Amway?
Way to doge the point. Harry is a wizard whether or not there’s a Hogwarts or he ever goes. Harry, like all the other wizards, has innate powers, innate power that were the story written by Christ Claremont would have been called “mutant powers” but instead it was written by Jo Rowling so she called them “magic powers”. The powers are there invitation or no, he can learn to use them invitation or no.
Then of course there’s the whole lie that one must be “invited” into occult groups. You’ve seen the books in the book store, people can learn all about the occult without ever getting any invitation, then they can make their own group.