Well I'm sure they're gonna figure it out when they look at Flash's book, or log onto any website, or see some old newspapers, or magazines, or all the stuff with Peter Parker's face plastered all over it

This will have been mentioned before in this thread I'm sure, but just saying. It's just movie logic you have to go by I guess, with how the spell worked.
Also during the time he was public enemy number one despite no evidence, I like how we see none of the Avengers came out to defend him. Like Sam Wilson "I'm the new Captain America and I know this kid and he'd never do anything like that and we literally both helped save the world"
But again, I know it's just a movie.