Stranger,
In my experience with the .303 cartridge, various marks of Lee-Enfield rifles and carbines, and a good number of similar military turnbolt designs I can say that the British combination can be as accurate as any Mauser-derived longarm if you look out for several things.
First off, find a clean No. 4 Mk. 1 or 2. While the older No. 1 SMLE (the one with wood to the muzzle) is neat from a longer historical perspective, the No. 4's (with muzzles that stick-out from the wood) are almost always better shooters. Second, look down the bore with a flashlight in search of heavy pitting. That's from the old mercuric priming compounds. A little bit won't hurt you, especially if you use 175-grain or heavier bullets. I've found the short 150's popular for reloading her in the States are easier to affect by a mildly pitted bore. Third, make certain the muzzle is free of any chips or dings that go anywhere near the rifling. They will kill your accuracy potential.
As for ballistics the .303 cartridge is more capable than the .300 Savage because of its heavy bullet abilities compared to the smaller American cartridge. With the old 215-grain loads available from Remington and CIL the performance against elk and moose is something to be seen to be believed. For lighter game we now have enhanced loads like the Hornady Light Magnum using 150's. Handloaders have several bullets that can do the job on just about any North American game. Stick to heavier bullets and the penetration capabilities of the .303 far outweigh the high velocities of the latest wonder short magnums. The .303 will be a great venison getter.
As for the Gibbs Quest, I'd say pass. The short barrel combined with open sights makes for poor accuracy under most circumstances, the American Rifleman test included. Stick to a full-length No. 4 and you'll do fine.
Please feel free to use the message feature if you have any other questions on these fine English rifles.