That is a super dumb mistake. There are also producers who are not members of the League.
A couple of nitpicks though, but they have no substantive effect on the situation. First, ATG is also a theatre owner. Second, the landlords have lots of contracts with unions, but do not directly contract with AEA (or several others) except when they also produce. AEA has a collective bargaining agreement with the League but contracts with each production. AEA has exclusive jurisdiction over Broadway houses, which are delineated in the agreement, so no one can present a show in a Broadway house without AEA.