I think the most interesting thing with timeskips is that the length of timeskip is usually indicative of the story drive behind the new series.
Your 'ok we're just skipping a few years' ones are essentially 'more of the same please' driven- even extending to sequels like Incredibles 2 where the extreme gap between films meant a lot of people were a bit disappointed to find it was going to be 'basically the next week' instead of having a similar in-universe time gap.
Paradoxically, the 'extremely large time skip' examples are also often a desire for 'more of the same please' but where it's a useful way of being able to just do it with a new cast and no strict ties- jumping a century later means our heroes can have won a period of peace and resolution even if the threats just re-emerge later.
I think the 20-70 year time skips are the ones which seem most interested in doing a 'same world, new stories', possibly because it's driven as much by 'what happens when our characters are older' as by 'something similar please'- both the Spidergirl comics allowing us to see Peter Parker as a retired superhero and father, and Legend of Korra giving us a timeskip to show how the changes of the original series of the world have led to a lot of political and cultural fall out really seem driven by that idea.
The TNG timeskip is a clever one because it came bundled with "and we are not mentioning X, Y, or Z, this is about NEW STUFF" in the bible that mostly held for several years - forcing writers to actually do new stuff instead of "hey, remember the Gorn?" (This may, as Lance Parkin's argued, partly be Roddenberry trying to severe the concept of Star Trek from Nimoy, Spock, and the Vulcans, which was the best thing for Trek's longetivity even if it was because the two men didn't like each other)