Yes, I've always field dressed mine as quickly as possible to help cool it down. Maybe it's just special deer I shoot but even ones that have run a little after putting one through the lungs I've never had one be tough or gamey. But yes dropping them where they stand is always preferred....also less work for me.
I'm far from a pro as well and also way more OCD than any processor I know so I always make it cleaner and better tasting than any processor I know.....plus it costs way to much to get one cut up around these parts.