Someone mentioned overcrowding in the morning. This may be at least part of the answer. IIRC, it is legitimate for a lounge to deny access to people with an access entitlement based solely on OW status, if the lounge is at capacity, or soon will be. After all, in the absence of being able to do that, the outcome would be that the lounge operator would have to deny access to people with paid memberships to the lounge, and/or premium-cabin passengers on their own carrier. If you had to make a choice, which would you choose? If this is indeed the reason, then better communication could go a long way in solving the problem, along with allowing access during non-peak hours.

Considering how new the lounges are at LAX, it could be considered poor planning if they routinely reach capacity. But, the change in terminal access rules for passengers at LAX, as well as AA operating domestic flights out of TBIT, may be even newer than the lounges, or at least their planning, and they're having to react to the changing situation.