I use the generators as a starting point because my mind doesn’t work well with blank sheets of paper, and the amount of time I fuss over it depends on the size of the character’s role. If I’m filling out a family tree, next to none. But if the character appears in the story with any appreciable role, I do spend time on them. The generators I deal with typically give lists, so it might be 50 names, and I’ll scan the list to see if any of them works, if not, I hit the “refresh” button. For characters with significant story time, I do try to make sure that the names are phonetically different, where possible, because names can mash together to the reader if you’ve got lots of “Jaimie” and “Jaden” and “Joe” and “Joey” and “Josephine” and “Jarred” and “Jefferey” and “Julian”, etc….
I know some people might dis on random generators, but to me, those are just suggestions, not like I’m saying “the next roll of the dice is...”, because it is my character I’m creating. And, as I work with characters, writing them, I’ll frequently change them anyway, adapting them to the circumstances, and how they write out on the story. (As I’m building a world, my DB is a couple of thousand characters, some more thoughtfully considered than others.)