I think I have found the distinctive difference when the application doesn't start for me.
Diia somehow detects that it was not installed from Play Store (though the Play Services and Store are installed). I use Aurora store. So it is sideloaded.
I have took spare smartphone with regular android and performed few tests. I may have Play Services and Store disabled post-installation, but as soon as I install from Aurora, Diia breaks. From Play Store - works.
The problem is that I don't plan to give Play Store an Internet access and I do not plan to create any new Google accounts to enable installation from Play Store.
I will try to find a workaround, read etc https://discuss.grapheneos.org/d/15666-some-apps-now-refuse-working-if-not-installed-by-play-store
If someone already knows how to trick any application to not complain it wasn't installed from Play Store - I will be thankful.
I have obviously tried protection modes - no luck.