My OS 9 machines (B&W G3 with G4 and PM 8600) are connected to a 5K display using a Gefen Dual Link DVI to DisplayPort adapter but they don't have GPUs that support more than 1920x1200 or 1600x1200. I do have a 7800 GT from a Quad G5 which can do 1440p60 or 4K30 but I haven't tried it in a PCI slot yet and it wouldn't have 2D or 3D acceleration in OS 9 if it could work. It would be for OS X Tiger or Leopard.

What mode is the graphics card trying to use when the display goes black?

I have no idea.

Some things to try:
- A modern display that isn't limited to 1920x1080 60Hz. I have a Acer XV273K 4K display which can accept anything up to 5K60 or 8K30 via DisplayPort. Not sure about HDMI/DVI though.

G4 Mini is limited to 1920x1200 on DVI and 1920x1080 on VGA. That's about it.

- Connect two displays - one display that is known to work so that you can see what timing is being attempted on the display that doesn't work.

There is no provision to connect two displays to G4 Mini.

- If you can't get two displays, then maybe a screen sharing app, so you can use another Mac to see what's happening when the display goes black on the first Mac.

Never used anything like that in OS9 and have no idea if such app exists.

These three are beyond the point

- A custom EDID adapter like Dr HDMI 8K.
- A scan converter that can take an input and convert it to a different timing that is acceptable by the display.
- A HDMI capture device (USB or PCIe) - maybe it can accept what ever is being output.

- SwitchRes - I don't remember if the OS 9 version has custom timing options? Maybe it can at least disable modes that don't work with your display.

It doesn't help, because computer freezes at the moment when ATI extensions load!

But, its not about my particular display, it is about latest version of ATI driver not working with ANY display.

There's nothing wrong with using smaller displays. I personally, despite having several modern 24", find myself more comfortable using 17"-er with my Mini.
Unlike in the old days, where 17" Apple's Trinitron was my main display and PM9600 or G4 DA were my main drivers and I had icons on the desktop in two layers on top of each other, I simply do not have enough stuff to put on a 24" real estate anymore.
Sure, some audio editors/sequencers will profit from larger screens, but even then I find that old Cubase looks a bit odd on a big screen.

But, there are several problems with older displays:
1. Fading CCFL backlight that also changes in color.
2. Not so optimal viewing angles of TN panels.
3. Slow response times

This question intrigued me all the time, so I decided to run a poll. Spill the beans, guys 

Well to be fair, os9 as daily driver in 2024 isnt really an option

We would profit if we enter a mindset where its just a hobby rather
Than for professional use

I also have to strongly disagree! My 9-machines are still my main computers, and wouldn´t we have this ugly TLS-situation I would never boot into Linux at all.

So, no; "hobby" is the opposite of what I use Mac OS 9 for, and I would apreciate if people would benevolently respect that there are still others around who do not do any hobby or retro or just-for-fun with Mac OS 9. Of course I know that my situation (no smartphones around, no interrest in any "social"-media at all, etc.) is very special, and a small minority is acting like me. But we are in existance, and we have very good reasons.