It’ll support 240 hz but your 60 year old eyes won’t.
The original vanilla client can run staggeringly high refresh rates. The modern client and vanilla client at their base are very similar.
The biggest difference between modern and vanilla are game design elements and how data is handled network side.
This response is extremely generalized because it kinda had to be.
There’s a 100 fps cap by default, but you can change/disable it in the advanced options.
I run it at 120hz which matches my 34"widescreen 1440p Alienware.
So yes it has nothing to do with anything but your graphic card and monitor.
Good news, it is a mostly smooth 240 so far. It dips to 170 or so in overcrowded areas, but that is mostly fine, especially with gsync.