No. If Captain America exists, he'd still be created by the private sector.
I think what happens with Superman depends on who the hero is. If the hero is interesting enough for the story to catch on, then other creators would make characters inspired by both the hero and Superman.
My favorite idea is a reporter named Clark Kent fighting against Superman. I also like the idea of a spy fighting against Superman, or even a spy pretending to be a reporter fighting against Superman. If the reporter story catches on, there would be more stories about heroic reporters fighting against weird villains. If the spy story catches on, there'd be more spy stories. If soldiers fight Superman, you'll see soldiers discovering supervillains in their wars.
And whatever happens, Superman probably won't be the only villain. More villains will pop up. But it's a possibility that the story would be more like Flash Gordon, where it's usually Flash vs. Ming the Merciless.
This could inspire Bob Kane and Bill Finger to create a hero fighting against The Batman. Some of the other heroes we know will appear as villains, but probably not all of them. Some may appear, but in different genres. Green Lantern might fit neatly into a magician genre, like Zatara, or might be exactly the same, but be considered to be a character of the magician genre. Of course, the genres may blend together if "Clark Kent" ends up fighting a wide variety of fantastic characters, including aliens and magicians. In that case, Green Lantern and Zatara would fit right into the new fantastic hero genre.
I think if there's a big action reporter hero fighting monsters and supervillains, Fawcett might still create Billy Batson and have him fight Dr. Sivanna, Mr. Mind, and even Captain Nazi. Maybe he'd fight a wizard as well.
The possibilities are amazing. I can't think of any possibility that wouldn't be cool. I think there would be mostly heroes without powers and a few magician heroes. Eventually, there will be heroes with special powers who are not magicians, but there would be a larger number of heroes without powers fighting villains with powers instead of the other way around, which I think makes things more interesting, because the writers would have to focus on the heroes' personalities, and clever ways for the heroes to win.
And yes, I think Captain America might be one of the first superheroes, but I'd prefer it if he wasn't one, and Timely published a story about an unusually talented soldier with a big flag on his uniform instead. I'd rather have Prince Namor be one of the first superheroes, staying pretty much as is, and untrusted because of his powers. And I think The Human Torch would make a pretty scary villain. Ideally, I'd have no actual "superheroes" until the silver age, with Stan Lee publishing the Fantastic Four, and having them untrusted like Spider-Man and the X-Men. But I'd like to hear other people's thoughts.
