You're looking for a deer gun on a budget. Might help if you specify rifle, shotgun, or ml.
Personally, I wouldn't buy the combo from cabellas. I have heard more bad than good about those rifles and you can get a decent used rifle for the same money that will out last those cheap things, esp. if you're lookin for 30-06 because lets face it, the 96 is more than likely the most popular hunting cartridge ever. So my first recommendation is to go for a used bolt action or semi auto.

My second recommendation would be to go for good used lever action in 30-30 or something bigger. again you'll have quick repeat round.

As for getting the good old NEF single shot rifles, I would recommend it. Here's why, although they're great rifles and I have had my 243 NEF single shot for years and have killed many deer with it, if you are looking for a single shot rifle for cheap then my recommendation would be to look at the inline muzzle loaders like the optimas and wolf combos. You can get one of these and hunt two seasons in most states. They're more than capable if tuned up right at taking deer at 200 yards. If you want a good deal then that would be my recommendation.

As for the 870 being highly inaccurate, your cousin is wrong. I slapped on a red dot scope on mine and its good out to 100 yards which is more than enough.

Those are my suggestions, I hope that helps.