As noted in the paper I linked to, display framerate did not affect the timing precision of their participants, from 120Hz down to 24Hz. So it's more about feeling it rather than seeing it.
If you can be accurate to below 5ms, an additional millisecond of input lag makes a big difference.
That paper measures our margin of error when syncing to a perfectly predictable rhythm, not our ability to detect system latency. Applying that to an FPS doesn't really work for two reasons. First, enemies are erratic, meaning you are mostly relying on your reactive visual-motor pipeline, which is bottlenecked by ~150ms anyway. Second, even for purely predictive shots, your brain subconsciously calibrates out fixed hardware lag.
I'll concede that pure prediction absolutely exists in FPS games-like tracking a falling player with the LG in Quake 3 or landing a predictive mid-air rocket. In such cases smoothness of high FPS helps, but adding a 1ms drawing delay to those 500FPS (what the article is about) won't be noticeable at all because your brain naturally calibrates out constant hardware delay.
If you can be accurate to below 5ms, an additional millisecond of input lag makes a big difference.