Something that has always bugged me about alternative history worlds is not the premise, or the events that might happen where the timeline splits off. It is more that people we know exist after the split. Given how biology works, ending up with the same people after a split in the timeline (with consequences) becomes almost impossible. When making any decision to split a timeline, one must tread, and know the time carefully. Knowing what famous people, born after the split would do in that world seems a bit ridiculous, but I guess that would be a measure of the "hardness" of the genre. For sake of discussion- What are the best alternative history stories you've read, and did they have characters that we would know, born after a split? Did they explore the split in detail? Does this desire to see science followed bug you as well? What's your favorite split off, either already done as a novel, or as one you would ponder doing yourself for a story?
Harry Turtledove's World War series and the follow on Colonization series addresses what you've discussed. While I understand what you're saying, the fact that individuals (scientists/generals/politicians/inventors etc.) were born/raised in the alternate history--after the change in history's course, but still rose to power/influence and had some of the same views/characteristics/personality, etc. it didn't bother me. It's an anchor. It's a tool for a theme or perspective.
I'd say a writer shouldn't care about that, except of course if they want to make it the point of the story. The best story with alternative history I can think of is Watchmen(which dealt with the effect superheroes would have had for history), although in it the split occurred quite recently(around 1939/1940 I think) so it didn't have to deal with the issue you raised. It's like in chaos theory: the changes are cumulative and the earlier in time you set the split, the more radical the changes should be. I once read a non-fiction book on alternative history where a bunch of historians wondered different scenarios: what if Jesus hadn't been crucified, what if the atomic bomb hadn't been used against Japan, what if Lenin had died before the Russian Revolution. Fascinating stuff and I have sometimes thought about writing something like that. I wouldn't write it in the broad perspective but instead from the point of view of individual people. I had an idea of a world where the Tunguska incident in 1908 would have been a global catastrophe.
Do you want to make a statement or are you writing something that's just fun? For example, there's a book called Guns of the South (also by Turtledove) about South Africans who time travel back to the U.S. Civil War and give the Confederates AK-47s. The South ends up winning and IIRC, Turtledove wraps the thing up in a few paragraphs, with the South ending Slavery and reuniting with the North shortly thereafter. The reader can't be so daft as to take such a story seriously. It's a fun experiment in science fiction and not a lot more. And as Makeshift said, the story's much easier to manage when written from the perspective of individual characters.
