I don't think the NFL owns Arena football, but I could be wrong. Whether it's a separate "code" or not is a matter of opinion.
In the late 80s or early 90s one of the US sports networks, probably ESPN, started showing Aussie Rules, and it became well-known. I remember seeing a game when the comp was called the VFL, which I think preceded the AFL, so it was a while ago. I don't know if the Aussie rules clubs in the US started as a result of that TV exposure, or if they were already there, started by ex-pat Aussies. I do think the interest in Rules was kind of a fad, and it's died down now.
I don't know why RL can't get on TV here and receive a similar boost in awareness and popularity. But I think it would be confusing to the average US sports fan, who wouldn't know that there are 2 different codes of rugby. (There are lots of ex-pats who do know, of course, but I'm talking about the average US sports fan.) Also, Union has a much bigger grass roots in the US, but it is still a very marginal sport. When the US team won its first game in the Union WC, the coach was asked if it would be a big boost to the sport back home. He said to be honest he didn't think most people would notice. You can say that Americans would like league better than union if they got to know it, and that may well be (because it's easier to understand, and closer to gridiron than union is), but I don't know if it's ever going to happen - not for a long time anyway.
Maybe if league had a different name it could make a splash in the US as a brand new sport, without being confused with rugby union. Something like Tackleball, perhaps.