Now, of course, any Vulcans may have a drastically changed present, some may never be born, etc. But, someone from Earth, such as Captain Picard, would have to have his parents meeting be affected by Vulcan still being around, such as a mission they went on together in relation to the Vulcans, where they met or a Vulcan who was killed in the destruction of Vulcan saved one of their lives in the Prime Time line, etc.
It doesn't need to be so direct. Upon everybody simply hearing about Vulcan being destroyed, they would have stopped what they were doing, changed their plans, and changed their thoughts about many things. That alone might not change Picard's grandpa meeting up with his grandma, but the chances of Picard's grandpa having the exact same sperm reach the exact same egg at just the right time is astronomically improbable. Add in his other grandparents, his parents, and the odds of all of them uniting, and the probability is about as close to zero as you could get.
Now, Star Trek has had its share of improbability when it comes to this. First Contact is definitely a good example. The Borg go back and change things, and the Enterprise goes back to minimize their damage. But the changes still happened. When the Enterprise returns to the future, it's highly unlikely that it would be the same future they remember. Usually that just happens to not impede the narrative or mindfuck the audience. But the next gen crews are all unnecessary. They don't need to be there this time around for any reason. It's not helping any narrative in any way.