The mid-credits scene was a great nod to classic Spidey. Being perceived as a menace by the public is part and parcel of Spider-man, JJJ having first appeared in Spider-man #1.
That said, I don’t see how what we saw in that scene is supposed to be a threat to this iteration of Spider-man.
First off, by the end of the movie everyone relevant to Peter already knows he’s Spider-man. So it’s not like it’s directly affecting his personal life.
Second, if Peter ever becomes afraid his nearest and dearest are under threat, all he has to do is call Happy - or Fury - and a quinjet will appear within ten minutes to whisk them off to a five star hotel, where they will be safe and protected in the lap of luxury.
Third, you’d think that both Fury’s counterintelligence guys and Happy’s PR team are pretty damn good at spinning the news, and they certainly have the resources to do it. And there’s no reason for either of them to back up Mysterio’s version of events rather than Peter’s.
Fourth, even if a combined Fury/Happy PR blitz can’t convince the public that Peter is OK, he can always go work for Stark Industries.
Basically the dilemma that classic Spidey faced - that he was an ordinary guy with no resources at all facing a media mogul - doesn’t apply to this Peter Parker. He’s a 1%er now. Worrying about what the press says is for little people.