I don't think it "shits on" a previous series to have a new character show up as the long-lost and never-before-heard-of relative of a character from said series. And a character could have relatives we've never heard of, that's true.
Nonetheless, I still don't like it when that happens. I see it as a storytelling crutch. To me, it comes off as if the writers/producers/whatever are afraid that the new character isn't engaging enough in their own right, so they have to be "propped up" by a popular predecessor. Why should I care about this crazy Vulcan who wants to find God? Oh, but wait, he's Spock's brother -- surely you have to care about Spock's brother! Why should I care about this character from a new series, who gets into major trouble in the first hour after I meet her? Oh, but she's Spock's foster sister!
It's lazy. It's a crutch. And if anything, it sells the new characters short. The Final Frontier isn't a great movie, but Sybok is a decent character. And Michael Burnham is interesting in ways that don't have anything to do with her relationship with Spock. (Some of those are connected to her Vulcan upbringing, true -- but any Vulcan family could supply that.)