How to find color profile for your monitor
I have the PA248Q too, and really like it = have found it to be an excellent monitor. This comment doesn't address your question directly but may help add to your thinking about other modes for using the monitor for future applications. This may include your web-browser depending on which one and version you use. So which one to use in the end will depend on what non color-managed applications you use to look at images. However, non color-managed applications will still show colors wrong and they will be MORE wrong in default mode than in sRGB mode.
If you do calibrate your display, you can revert back to the default mode (called Native on some monitors) and then all color-management-aware application will be able to:
The truly odd thing is that those particular colors are kind of random until you calibrate your display! This is still more or less within the bounds. Once in sRGB mode your monitor may not be able to show colors which are outside of sRGB color-space which is why sRGB is not the default mode. I resolved the issue by creating and adding lines to nf, ignoring whether stuff is working via xrandr.
Keep in mind that this mode is not calibrated, so your sRGB colors will be different from other sRGB colors.