Can you tell for certain that a model wrote it?
No, and be wary of anything claiming otherwise — detectors that promise to identify machine-written text are unreliable enough that schools have stopped relying on them. That is why the question worth asking is not “did software write this” but “is this a real enquiry”, which is answerable from what is being asked and what is being claimed.
Some of these are sent by real people, though.
Quite a lot of them, and it makes no difference to the decision. A person pasting a generated pitch into your form is sending you the same message for the same reason. Judging the content rather than the sender is what makes that distinction stop mattering.
If I filter these, will I lose a real enquiry?
It is the right worry and the honest answer is that any filter can be wrong. What matters is whether you find out: a CAPTCHA’s mistakes leave no record and no name, while a held message stays in a list with the reason it was held. Start in dry run — a week where nothing is held at all and you simply read what would have been.