Thanks for updating! FWIW, it sounds to me like you handled that call really well. A few things sound odd to me - that the supervisor would say she's worried you'd badmouth the agency. She could simply have said, she's concerned there could be a conflict. Without the accusing tone, for something that you haven't actually done (badmouthing the agency)...And if they're not doing their job right, well it's not your problem, it's THEIR problem. You'd be justified in expressing your opinion to other parents.
Admitting the workers is burned out on the case seems like a good thing but then why give an excuse? That's irrelevant to you and IMO shouldn't be brought up to you, as a client. If you aren't doing any of those things that are listed in the client agreement as a reason to terminate services (failure to show up for appointments, etc) then the fact that the worker has burnout is NOT a reflection on you.
I'm glad you'll get the chance to meet the new worker face to face and get a read on her. I hope it works out for you and DD!