That's the rub; it is in fact a pun and a joke. It's a joke rooted in ideas that they're reinforcing. Jokes can be quite powerful in shaping norms simply because people think they're harmless. Humor works to disarm tension one might otherwise feel when having a normative viewpoint presented to them.
Minstrel shows were literally funny to both proud racists and people who simply weren't the targets of racism. They would find humor in black caricatures behaving like they weren't inherently inferior to white people, because the prevailing perspective was that they were inferior. People who didn't understand the humor could even laugh along, not knowing explicitly what they were laughing at, but absorbing the viewpoints all the same.
Not every example is so obvious, especially when crypto fascists are involved. The alt-right were masters at double layered jokes that work on both an offensive and non offensive level. I spent a lot of my teen years lurking in alt-right spaces on reddit, and it was enlightening how often some people missed the fact that they were in a fascist space. They were simply there for the normie humor; the jokes where the punchline wasn't just racism, but puns, social cringe, and dark humor.
So yes, the simplest answer isn't that the American Eagle advert accidentally looks like alt right humor.