I apprenticed under my father to learn machining and tool and die making.
Gunsmithing came naturally after that, and instead of being shackled to "oh dang I need to order a part" I can often fabricate the part faster and with better materials than the original or replacement.
The specialty schools only teach the little tips and tricks specific to gunsmithing (which you can learn from books, youtube or talking with old smiths), as well as give you a foot in the door when dealing with distributors and such. Having actual skill when it comes to gunsmithing is a matter of practice, which you can get at home or at a school.
The only benefit I could see with a trade school is you could get scholarships and such to help pay for it, or you can just take machining classes at your local community college.