That’s not too true, response time is nice but not all that important.
Let’s just look at all the delays in the entire chain from input to output. Your mouse already has some delay, even a cable mouse has a few milliseconds delay, most gaming mice have around ~10ms delay, some a bit less, some a bit more. (And even if a mouse would have no delay somehow, USB and polling rate alone can already add up to 2ms of delay.). Then we have the input lag from the game:
I play with 3000mhzRam, an I79700K, and a gtx1070 on all low settings, 103pov, and 75% render scale. I have a delay of 7ms, especially on better graphic settings and a less good PC this can be much higher through. On default setting my latency was 13ms.
So now from input to output, it’s already at least 18ms delay, even with a 1ms monitor. The difference between a 1ms and 4ms monitor (which is the highest I sometimes see with gaming monitors) is from 18 to 21ms. Now consider the reaction time of your body, which is for a physical response of at least 150ms if you are super fast. To be fair you can’t really add reaction speed into it like that, I wanted to give it as a comparison through. In addition, the 60frames alone already have an average delay of 8.3ms.
Even if we remove reaction speed and ignore the reaction speed of the mouse, the latency of the game and framerate is already more than quintupled compared to the different delay of gaming monitors.
Now compare the delay alone from a 240hz monitor to a 60hz monitor, there are other benefits, I just discuss the delay here. The max. delay of a 60hz monitor is 16.7ms, the max delay of a 240hz monitor is 4.17ms. This is only the maximum delay, the average delay of 60hz is 8.3ms and from 240hz 2.09ms.
That’s an average difference of 6.2 milliseconds.
So even if latency would be the most important thing:
I (besides some extremely rare exceptions) have never seen a 24-27inch sized gaming monitor with more than 4ms latency, which means the latency difference from going from 60hz to 240hz is more than doubled compared to the pure latency the monitor itself has.
The average delay of 144hz is 3.5ms, so the difference of 60hz to 144hz is already 4.8ms on average.
As long as your computer is capable of it, upgrading from 60hz is always more worth it than purely upgrading the latency itself. Of cause both is nice, but framerate is more important