True, but don't you hate the wash out colors from a TN panel? That is exactly the same one that I have now, but larger. Maybe I should just save money and get a 120+ Hz IPS panel. I know a good one may cost $500-$700.
So i'm doing this same exercise right now. I'm currently using fairly nice IPS 1080p 60hz panels in 5760x1080 surround configuration, I spent a week gaming on my Samsung KS8000 at 4k, and I've spent time playing on my friends 144hz Freesync panel. I found the better colors, deeper blacks and higher resolution from my KS8000 (quantum dot + high end VA panel with local backlight dimming, very high contrast ratio and HDR capable) were much less impactful than going from 60hz to 144hz for gaming.
In gaming it all is a game world so the color accuracy of the monitor just doesn't really make a difference in my mind, because its all far enough from reality anyways. It doesn't add anything for me. I was playing Total Warhammer II, I looked at the deep greens and shadows on trees in the jungle and sure it looked great at high res with that color reproduction, but it was noticeably choppy even on my 1080 ti at 2050 mhz. And the color difference was totally outweighed by having to turn down settings, losing SSAO looked way worse than switching back to the 1080p monitor. I actually did worse in PUBG because the deeper blacks made hiding in shadows easier for enemies compared to the lower contrast on my 1080p IPS monitors. The 4x resolution helped a little in terms of noticing enemy movement but was more than outweighed by the deep blacks hiding them better.
Having tried all 3, I would say for any type of 3d gaming even turned based like Total War campaign mode the smoothness and dramatically lowered input latency of high refresh rate is absolutely the bigger improvement. Just look at this article to see how much input latency you can cut off going to a fast refresh panel:
https://danluu.com/input-lag/
I'm going to meet in the middle with a 1440p @ 144hz panel and get a little of both
In terms of actual impact on the game, I'd rate display qualities in this order of importance:
Resolution until you hit 1080p > Ultrawide (adds real FoV) > refresh rate > resolution over 1080 > contrast ratio > color reproduction.
All this to say a decent TN at higher refresh is fine, if you can afford IPS its better but I personally wouldnt pay much for that privilege unless I did something else that required color accuracy like photo editing. I can't comment on the 165hz vs 240hz vs 144hz thing. There are some TNs that can hit 240 but I dont know if that is noticeable, usually you couldnt even push that high of FPS even if you want to on anything modern due to CPU bottlenecks.