In my experience AA doesn't tie up too neatly when you are a OW elite on another OW carrier. I do about 2/3 of my flying on AA but am a LAN Sapphire.

I find that although my FF number appears on my boarding pass, status does NOT. The only time I have really pulled them up on it was in HNL when my FF number didn't appear on my boarding pass.

I asked them to make sure it was added to my reservation, the woman checked and at that point manually stamped my boarding pass with "Gold Lane"

So it seems that in the system the status appears but it does not show on the boarding pass, not much use if you want to use AA preferential boarding because your boarding pass doesn't show OW elite status.

That said AA doesn't really extend many benefits to other OW elites, for example no access to blocked out seating as an AA elite would have on LAN. As a Sapphire it would be handier for getting into their lounges if they could print status on the BP