100% agree in principle, but in reality how many GP's know which of their patients hold firearms ?
I dare bet that any enquiry from the Police to a GP is fielded by an 'office girl' who may check the records and declare the applicant 'safe'. The Doctor is probably totally unaware of either the enquiry or the 'patient' holding a firearm.
In the last 15 years I've been to my Doctors surgery only once, and that was for an 'over 50's well man check' (blood pressure, diabetes, weight, chloresterol, etc) so he would not have a clue about my mental state.