And because I receive all those useless e-mails as well, about things I know, I don't want to add to it by having BA e-mail me about things my TA and I have already discussed, and I have already received e-mails about from the TA. I presume YMMV.
If the communication breaks down, such that something happens as the OP described, then it's not BA's fault. It's entirely between the traveller and the TA. We should not be expecting BA to do work which the TA is getting paid to do.