Thanks to all for their comments, seems that I am finally not alone having the issue with this browser. It is important that if you experience this you star or better even comment on that bug report I linked earlier, a Google account should be enough to log into that system:
This increases our chances of Chromium devs looking into this, so far there's been basically zero activity.
I also opened a Google support case for this pointing them to the bug report, it's been even escalated to higher levels of support (at least they claim so), but no real outcome there so far.
In the meantime I was trying to get accustomed to the problem and see if I can get used to it: played on Stadia for several hours on Dec 23, felt like it could finally be OK at first, but by the end of the day and the next day I paid the price: terrible headache, feelings of nausea etc., partially ruined my Christmas... Chrome is unusable for me if this does not get resolved, that's for sure.
Additionally, I have some news, but it's not looking good. I spent half of my yesterday setting up and compiling my own Chrome from sources on Linux trying to disable both of the dithering options the bugs I linked earlier speak about. And I was successful in the end: my chrome build is showing obvious color banding in various tests (unlike the official build), so I apparently achieved the dithering removal. But the effect of this custom build on my eyes is just the same. Therefore it must be some different form of dithering or it is indeed the color profile handling of Chrome doing this (so @Seagull might have been right, my apologies sir for doubting what you said). I am at a loss. I will follow @Gurm 's advice of trying out Chrome v68 if I can find it somewhere to see if it is OK so that I am able to point Google in the right direction as my original assumption was apparently wrong. There was color management implemented in Chrome some 1-2 years back so that could be it, too. But even at the very beginning of discovering this issue (some 3-4 weeks back) I was testing various options available under the force-color-profile flag to no avail.
@Lauda89 It definitely has nothing to do with fonts as staring at pictures and playing games on Stadia causes the issue as well.