Keep working on it, Martin. It becomes second nature. True, after a day of walking (and sometimes a long drive home) cleaning the game is a bit of a pain, but it gets easier with practice.
Tip: Gerber's game shears work pretty darn well. Two pair of those and a Rapala folding filet knife always ride in my hunting bag.
Tip 2: Re the second set of shears: When the other guy says, "Gee, forgot my shears ... I'll just watch" hand him the spares. And if he says, "I don''t know how" ... hand 'em the spares and tell him, "Watch and learn."
Tip 3: Be careful trimming your game if you're very tired ... Was tidying up a pheasant under cool water just prior to freezer-bagging one evening and darn near cut myself to the knuckle.
Observation: There's a little bird that beats 'em all for scrumptiousness: bobwhite quail.