Based on whirlpool wiki plus a 1000 other charts and graphs that you can google if you like
Here is the 7850 vs the 560ti.Both in OP's price range @ $250
http://www.anandtech.com/bench/Product/549?vs=547
7850 comes ahead in almost all games. Thus best bang for buck ATM. I think this will change very soon though when we see the 660 and 670
while it shows that the stock cards at release, the card i recommended isn't a stock card, and makes up for what the stock card lacks, (like 2 gb ram)
and while I think your right regarding the field changing come the mid-low level gtx 600's, there is alot of bang still left in the 500's, friends have bought the card i recommended, and have run BF3 at high res with no issues and it comes in near enough to budget, it is tried and tested and while it might not have 22nm chipsets it handles all the OP asked for and more,
also, using those charts, bf3 is better on a 560, pretty sure a card with the same ram as the 7850 (oh, like the card i linked to) would show a better result, and would best the 7850 in far more tests