I personally scrap a lot more brass than probably most anyone in this thread has. Right now I have over 900 pounds sitting waiting to be sorted and over 400 pounds of scrap!
Most recyclers won't accept spent cartridges- AT ALL- because of the possibility of live rounds being mixed in. About half the recycling places near me won't take cartridge brass at all. There are two who will take it but only if it is first burnt or shredded- they cite live rounds as the reason. There are two who will take it as is- one is a specialty alloys company (didn't blink an eye when I needed to get rid of some cadmium) and the other is a smelter who is paying $1.25/pound for cartridge brass.
The recyclers that DO take cartridge brass will take it either as "Cartridge Brass" at a slight discount over regular brass prices, or as "Yellow Brass" the same as plumbing fixtures, valve bodies, etc...
I have never, ever been asked to separate primers from cases, or to even remove nickel plated cases from the yellow brass cases. Brass itself is an alloy composed of zinc and copper- a bit of extra copper is no big deal. I dump all my spent primers in with the scrap brass. The spent primer is brass as well. The trace of nickel isn't a problem. The scrap classification "Cartridge brass" allows for trace amounts of nickel as well as contamination with residue from firing.
The reason the scrappers are probably saying you have to take out the primers is probably A:, they don't want live rounds, B:, they are trying to get it cheaper, C:, it's too much of a bother for them to deal with what is probably a small amount of scrap they don't see very often as compared to #2 copper which they get tons of, so they blow you off.
Aluminum 'Blazer' cases can't be recycled as aluminum without removing the brass primers. They go as mixed metal scrap. Honestly, my buyers don't even want them at all. So all the aluminum cases I wind up with get mixed with all the steel cases, the clips from smelted wheel weights, the dross buckets, old mower parts, appliance scrap, etc- and those get dumped in cars I take to the shredder, which is right now $240/ton.
I even save the bullet jackets from smelting range scrap and magnet-sort out the steel ones, then the copper jackets go for #2 copper at the smelter.