Sounds like you have a good bloodline and a great market for pups. Here, it's puppy ranch overload, the local (and very new) Humane Society can eventually get you your choice of breed.
No argument, an AR of either style isn't a common scabbard rifle. With the budget described, a bolt .308 or big Marlin lever would do. But that's not to ignore a future decision.
AR's in different calibers are quickly catching up as the most sold. 6.8 SPC is the front runner, and offers a whitetail or hog a one shot stop - from a 16" barrel. The large .458 or .500 is also good for tougher game out to 200 yards.
What the AR offers is a low recoil, high speed round with great accuracy, durabililty, and lots of options to kit out how it needs to be used. AR's are great hunting rifles and the biggest selling category these days, with most AR makers, including Remington, offering a version.
Just like all military rifles of the past, it's becoming a popular tool for good reasons. It may not fit the image of horsepacking hunters, but I imagine to the old boys still using flintlocks in the 1890's, a lever was as out of place, too.
It's just a matter of perspective.