A 120hz output refresh rate means little if the games are still 30-60 fps. And they will still be 30-60 fps.
A 120hz output refresh rate helps reducing lag even in 30 fps games and 60 fps games.
The refresh rate of a monitor has an impact on input lag. A 60Hz display, for example, will never have a visible input lag below 16.67ms, because that’s the amount of time which passes from one refresh to the next. A 120Hz display halves that time to 8.33ms... and you don't have to render 120 fps to benefit from that smaller lag.
With VRR there is even less input lag, since the rendered frames are sent instantly to the display when they are finished, not 16 or 8 ms later. If done properly, games with unlocked framerate can stay smooth if they stay above ~40 fps.