@Sarah977 I have had hotel maids walk in unannounced when I was getting out of the shower or if I had my Do Not Disturb sign up, so I guess I think this isn't quite as unique as others do. Was I happy that a hotel maid came in at a time they were not supposed to? No. Was I freaked out a bit by it? Of course. I can totally understand where the OP is coming from with regard to her daughter's safety.
As I said, its all in how the host handled the situation. If the host was defensive or nasty I would have escalated this. If the host was just explaining that this is a service provided to keep a place free of bugs and profusely apologized I would not have given it more thought. I guess I don't want to jump to the conclusion that the host was defensive and nasty? If he/she was, maybe the OP will come back and tell us.
All we have is "the host also knew I was traveling with a teen. The host was unable to perceive a safety issue for my daughter or even a liability issue for the Exterminator since he was unacompanied with a minor. The interaction did not end well with the host and I reported my concern to Airbnb." As a host myself I can't guarantee that maybe a census taker or a neighbor won't knock on the door with a teen present alone. Perhaps the host knew the exterminator was coming. Perhaps not. We don't know if the host tried to resolve this with the guest and the guest was too upset to accept a resolution, or if the host was just rude and the guest had no choice. I look at this as what would I do...apologize, offer some kind of discount, listen to what the guest's concerns were. And maybe that is what the host did, but it wasn't enough to make the guest feel comfortable. I would love to hear more about what happened.