What if Captain Anerica only made those P.S.A.'s for Peter Parker's school, Midtown?
We know that there are some really cringe Captain America psa's that are shown in the Tom Holland Spiderman movies, but one thing that i don't see mentioned enough is that the principal is the grandson of a Howling Commando.
Knowing how nostalgic for 'the old days' Cap is, he definitely would have searched out any of his old war buddies, which means he would have come across their families, too.
Imagine him meeting Principal Morita and them getting to talking about 'kids these days'. Steve remembers the old clips they used to show before movies and think (with rose coloured glasses on) about how well behaved kids were back then. He gets the idea to do a speech for the kids at Morita's school, thinking that it'll be motivational and they'll behave and be good people, or whatever.
The idea gets vetoed, but then someone comes up with the idea of filming a bunch of PSA's. They'll be inspiring the next generation, and they can be whipped out on multiple occasions! Steve feels a bit like a show monkey again, but he agrees to do the videos as a favour to his old friend's struggling grandson.
Problem is that word gets out about it, and so instead of being this sweet little thing for Morita, it becomes a big production and ends up going out to every single American high school.
Steve, however, is clueless, because he hasn't worked out all the ways the world has changed yet, and still believes that it's just being shown at Midtown High School.
Fast forward to Peter being picked up by Tony Stark, and somehow Tony has not come across these videos yet. So Peter shows him one of them and Tony goes and shows everyone the YouTube compilation of the videos, and this is how Steve finds out that the 'sweet little favour' he did and promptly forgot about is now mandatory watching in American schools.
If Steve's in the picture at this point, then he hadn't thought anything of Peter quoting one of them, because Peter goes to Midtown.