The problem so far
The problems around this thread seem to revolve around 2018 Mac Minis with 4K monitors not sending HDMI output / sending bad video output while starting up. And for some reason, the BIOS (or whatever is called) sends a resolution in the first screen that is not quite supported by 4K monitors or when detects a 4k monitor somehow manages to screw up the output on them, unless they are limited only to 30hz or similar.
Fixes that have failed
USB-C to DP: I have tried a USB-C to DisplayPort cable and it doesn't fix anything, but in my case the issue wasn't the HDMI port itself, but the incompatible resolution it sends, so the only difference on my monitor is that when in HDMI I get a message about incompatible mode and while in DP I just get an acknowledgement of receiving signal but no image.
Resetting NVRAM: I have also tried resetting the NVRAM, and kinda works one time, but only for that specific boot right after the reset, and sometimes with pink colors or similar bad output issues. .. and the next boot I will get exactly the same issues of not being able to get an output that my monitor can display, so it is not a sustainable fix (can't reset the NVRAM in every boot! and in the era of bluetooth, I need to dust off my USB keyboard to do so). Also the screen is on 4k without any scaling so using the mouse is annoying in that screen as well.
10.5.1: With 10.5.1 the issue remains the same... and as I have been reading that downgrading to 10.4 will not fix anything since something changed at a more fundamental level (BIOS?). So we cannot go really back to normal.
C'mon Apple take action!
Please, Apple, do your due dilligence and solve this issue. We need reliable setups. A recent Mac Mini with a Samsung 4k monitor (and according to other posts, also happened with Sony and LG TV/monitors) is not an extraordinary setup. And clearly we are many with the same issue, who are forced to use a spare monitor, or to blindly type their password. I don't think that anybody would spend >1k dollars to have such an unreliable fix.
Is there more to it?
Also, is it possible that there is a batch of mac minis with some hardware issue that is leading to this outcome?