How do NFA's make program design any better when converted DFA form can't remember original states?
[i]f the above DFA is implemented it surely finds a winning state on a "m"(move),but how does it decide whether it was right or left move?


In the example you gave, it can't. The nondeterministic transition from the start state occurs when a "move" is made. That's all you can possibly know; the machine can't determine whether the winning move was left or right, because those aren't allowed possibilities.