To be fair, cost, need, and usefulness of anything above the standardized 60 Hz minimum for different resolutions are the reasons why it took long for LCD monitors to go past 60 Hz.
The LCD panels were slow, compare that to a CRT with long decay phosphors. The settling time of new pixel value was long. But the image still needs refreshing at some rate or it starts to fade. The panels were meant for 60 Hz in every aspect, so they would not work at 30 Hz refresh rate. Also no computer had any standard formats that could go much below 60 Hz and there was no data structure in the monitor EDID that it could advertise support for formats below 60 Hz. So the best option is to make an LCD to look like a 60 Hz CRT as that is all that is needed for it to work without reinventing the whole ecosystem from the ground up.
If 60 Hz is enough so that the screen does not look like it is flickering, why go beyond 60 Hz? People may have used CRTs at higher than 60 Hz if it looks like it flickers at 60 Hz. Which is why even on a legacy IBM VGA, all the resolutions were 70 Hz except for the highest 640x480 which was 60 Hz, so in fact, if you ever used an LCD in DOS, it had to support a couple of 70 Hz formats anyway, I had a LCD which had a menu to select between 720 and 640 pixel modes depending on if you used it for text or graphics.
Also if you use higher rates, the display has to have a more expensive triple video ADC to sample at higher pixel clock. The display panel must also support the higher pixel clock rate on the digital bus.
Also all formats that are not the native panel resolution needs to go through a scaler chip, so again all the data that goes into and out of the scaler must have a sensible data bandwidth limit as it must always output the native panel resolution.
And again, if the pixels are slow anyway and can't change colour quickly enough to be useful beyond 60 Hz, using higher rate than standard minimum of 60 Hz for any resolution will just go to waste.
Only later when technology matured and there was a market for faster displays, they became available. Panels became faster, the analog monitor interfaces were replaced with faster digital interfaces, and other technology such as the interface and scaler chipsets got cheaper, and there was a market (gaming) which benefited from higher refresh rates, and also computers now had powerful graphics accelerators that could deliver 120 FPS content over 120 Hz resolutions.
And for the "Why 60 Hz, why not something else?" part, the general ecosystem with external displays had for a long time been using CRTs that have their history in TV screens and TVs worked at local mains frequency 50 Hz or 60 Hz, so CRT technology cheaply available for consumer products has always been around 60 Hz. During 90s when resolutions improved, and while the bandwidth can be used for higher refresh rates at lower resolutions, the CRT phosphors simply started to cause flicker if less that 60 Hz was used. So for most video cards and monitors, they used standardized formats and the highest supported resolution was only reaching the 60 Hz minimum required for usable operation. Standard video card drivers rarely allowed non-standard refresh rates below 60 Hz, and depending on the hardware, it might not be even possible to select custom pixel clocks or horizontal rates or lines per frame to support lower frame rates. And the highest possible refresh rate at some resolution is set by the pixel clock which can't exceed the limit of the chipset.
Even the EDID standard for PNP displays allows defining supported Established Timings starting from 60 Hz, the Standard Timing flags already being de-facto formats at 60 Hz or above. Custom format support could be defined by picking any within range of indicated Monitor Range Limits descriptor, or the monitor could say one or up to four Detailed Timing Descriptors but it does not mean that all video sources can configure itself to any requested timing format.