The skeleton of Weisman’s Peter and the Peter from TAS is the same even if we saw the latter start out in college. That has more to do with what point in the character’s life the writer wants to introduce the audience at. It doesn’t mean the things I mentioned (Peter being orphaned, getting bit in high school, losing Uncle Ben, etc.) didn’t happen. Lots of writers jump straight to Nightwing when covering Dick Grayson, but that doesn’t mean his backstory and his years as Robin aren’t there.
Being created as an icon and character aren’t mutually exclusive. I already brought up Kamala as an example, but a lot of Peter’s initial appeal was about representation too. In Peter’s case it was representation of young people and skinny men. The “young people” part is obvious (first independent teen superhero, first to not be a teen caricature, etc.). What I mean by the second part is that there weren’t many male superheroes that didn’t look like traditional masculine looks prior to Spider-Man. Peter in a way was proof that all men can be superheroes and not just a select minority of men that lift weights and are very athletic. A lot of the initial appeal of AF # 15 boils down to those two demographics finally feeling positively represented (That’s right, Peter is technically what the Alt-Right would call a “PC Character” . We just don’t think of him that way now because of how normalized people like him have become. We have real-life icons like James Dean to thank for this, but also fictional ones like Peter himself.)
You are right that Peter wouldn’t have had as big of a cultural impact had it not been for TASM, but I think that has more to do with social media and how fast information travels now. Also superheroes (Spider-Man especially) are a lot more engrained in our culture now to the point that The New York Times will write consistent articles on them.
The question then is why Miles has varied so significantly compared to other Marvel characters. I think it has to do with a lack of a clear vision behind Miles when he was conceived. With characters like Peter Parker and Kamala Khan, a lot of their struggles in their initial runs were based on their creators’ experiences growing up. Bendis was well-intentioned, and his overall run on Miles isn’t bad, but it was lacking that authenticity and personal connection to the character that the Lee/Ditko Spidey and Amanat/Wilson Ms. Marvel have.
So we were left with a Miles that was, to a huge extent, a blank canvas that people didn’t know what to do with. This became especially true after he started co-existing with Peter in 616, which meant that being the guy who took over from Peter wasn’t enough. Most writers at Marvel apparently didn’t even know what his new 616 origin was or if he still had memories from the Ultimate Universe. The comics are only getting around to addressing that now.
The impression I get is that ITSV was the turning point for Miles, and that everyone else is now stuck doing cleanup when they would rather just import the movie version into the mainstream as if it were Bruce Timm’s Mister Freeze. In this case however, it isn’t as easy to do that. Both the Miles in 616 and Insomniac’s game have a previous continuity that has to be addressed. For example, Saladin Ahmed’s Miles is closer to his movie version, but the book is also trying to clear up the continuity issues from Bendis. Insomniac’s Miles already looks less nerdy in the trailer than he did in the first game, but it remains to be seen to what extent they can do a 180.