Assign values to the non-king chess pieces as follows: Pawn=1, Knight=3, Bishop=3, Rook=5, Queen=9.
Now consider this variant that I will call Plus-3 Chess. Plus-3 Chess has exactly the same gameplay as ordinary chess, but a player wins if either a) he wins by ordinary chess rules, or b) it is her turn and the value of her remaining pieces exceeds the value of her opponents pieces by 3 or more.
For example, in plus-3 chess if the material is even and White captures a bishop and Black does not capture a piece in the immediately following move, then it becomes White's turn with a material advantage of 3 and White wins.
Has this variant been considered elsewhere? Does it have a fancier name? Is this game easy to solve? ( Weakly and/or Strongly? )
Now consider Plus-k Chess for k=4,5,etc. that are defined like plus-3 chess except they require the material advantage to have a value of k. It seems to me that plus-6 chess is almost the same as ordinary chess for nearly-matched players. But of course you are probably able to construct a position that wins in plus-6 chess that is losing under continuation in ordinary chess. So instead let's examine all Grand Master tournament games in the history of Grand Masters, and find the maximum material disadvantage that was overcome. Call that value K-1, then plus-K chess is "tournament-equivalent" to ordinary chess. So solving plus-K chess, while not strictly equivalent to solving chess proper, should be good enough for any human player. And should be easier.
