I'm not aware of any issues with TA bookings on other airlines. To be fair I'm only really familiar with AC's upgrade program (old paper certs and new "eupgrades"). There may be corner cases but I don't know of any general issues with TA bookings. I don't believe United or AA has any warnings about how tickets are issued in their docs but I haven't actually used either program.

I think this comes down to what the contractual relationship is between the airline and the agents. For some airlines the airline seems to still have rights to make all kinds of changes to the reservation or your agent can do some of those too but it depends on what the airline allows them to do. For BA apparently their contracts are much more all-or-nothing where the agent is supposed to be 100% responsible for the reservation unless BA takes 100% control. I suppose there are legal liability reasons why that might make sense from BA's point of view but it doesn't seem to concern other airlines. Or maybe it's just because that's traditionally been the relationship BA has with their agents.