I am going to be the turd in the punch bowl by saying this but.... You baited and placed a trap, and caught a stray cat. Why take it to the animal people? Why do you assume a cat needs rescuing, and why do you assume the shelter people are the solution. Assuming a cat needs "rescuing" was your first mistake. Taking it in to those people was number two. What did you expect? Let the damned thing out and on it's way and you would have avoided a lot of ass pain.
I disagree with your statement. If there's reason to think it's a stray and not a feral, there's good reason to think that bringing it to a shelter might reunite the cat with its family. OP also asked around the neighbouring properties to see if they were the owners.
Additionally--and maybe it's different where you are--adult feral cats are often caught, checked for diseases such as FIV and FeLV, neutered/spayed, and then released back into their colonies. Kittens and well-behaved adult cats are adopted out. This helps reduce the impact that ferals have on the native fauna by removing the ability for colonies to increase without having to euthanise the animals. Diseased cats are destroyed to stop the spread of disease to pet cats.
The neuter and release method also encourages people to bring the animals in as there is no guilt associated with doing so (the captor knows that the cat will not be destroyed). We have some overwhelming issues with introduced species here in Australia, so every move towards better conservation has a positive impact on reducing the harm done by these introduced species.
I understand where you're coming from and maybe the neuter and release programs aren't available in your country/location, but where I am it makes complete sense to do what OP did.