Does this host own the space? if so they probably just forgot the exterminator was coming. We have one that treats the outside of our cottage and sometimes they give us a very short notice window. We always try to reschedule if a guest is there or arriving, but its not always possible. They do not have to enter the space. Same with our landscapers.
If the host is renting, they may have no idea when the exterminator is scheduled as the landlord will have that arrangement. I actually lived in an apartment in Chicago years ago with this sort of arrangement The exterminator was the LL''s brother-in-law and he just showed up whenever he had a moment in his schedule. I would call the LL to verify he was supposed to be there. We had to move dishes from cabinets, etc so it would have helped to be prepared. There were many times I had to send him away and get ready for him to return.
You never have to allow anyone into your space. While I agree that it would have been best to give you a heads up, I honestly don't think this would be enough to ruin my stay. As long as the man did not come into the space while your daughter was alone but merely announced his presence, I certainly would not have elevated it to Airbnb unless the host was very rude or defensive. If the host was apologetic and there were not other issues, I would have just left some private feedback. But that is me.
Hosts do their best to make their guests comfortable, but we can't always control what goes on in the environment. The techniques used to address your concern would be the only area where a host might have some agency or impact.
That said, we would get more complaints if our cottage was not treated for various insects. As it is, people have lamented that our rural property is, in fact, in nature. Sometimes hotels with more strictly managed protocols are a better choice for certain guests.
