Were they? The Howling Commandos were knocking over HYDRA camps, and while that is the kind of highly specialized assignment, it doesn't veer into the type of black ops that the government want to deny.
The thing with Cap is this; he's this awesome, badass soldier who goes into battle wearing the flag. He's unusual and thus memorable, and
he would have been seen in action by hundreds if not thousands of troops, not only of the US armed forces but from other allied nations as well.
After the war, his legacy would be commemorated, if not by the government then by the troops he commanded, who would have no reason to just keep it to themselves. Books would be written about him and at some point, Hollywood would make a movie about him.
At the very least, he's as well known as Patton.
Of course, not everyone could recognize a picture of Patton, even if they know the name.