<Progressive Chess is a definite candidate ! Although it is not one of my favourite variants, it has the advantages of being already very popular, and to produce fast games (games >10 moves are an exception).
I would imagine that implementing multiple moves would be more expensive in coding than some other variants. But once done, we could gain a lot more : other popular games like Marseillais Chess or Double Move Chess. These are two different two-move variants. In Marseillais Chess, like in Progressive Chess, if you check with the first of your two moves, you lose the right to play the second one. In Double-Move Chess, you can win by checking on the first move and taking the king on the second.
Other excellent multi-move variants are Avalanche (after each of your moves you have to push an enemy pawn one square forward), and various asymmetrical games where one side has only 3 or 4 pawns against a whole army, but can play two moves at a turn.