Game has to process the input, but it also has to update the "world" (which might also involve separate processing like physics) and then also render it both visually and audio. With network and server updates in-between things get even more complex. Input to screen lag and latency is a hardcore topic. I've been diving into that on and off for the past few years. One thing that would be really sweet of hardware/OS/driver guys would be an info when the frame was displayed. There's no such thing yet available to my knowledge.