Problem solved.
It looks like something has changed either in Xcode, or in iOS 16 (I'm more to Xcode thing), but all in all, the size of images used in Widgets now matters. In iOS 15 and previous Xcode we were using 4 images of ~500 Kb each, and for some users in iOS 16 after building with Xcode 14 it started to cause memory issues, which turns widget into "placeholder mode". Strange, but it didn't affect iOS 14 and 15 users at all.
Reducing the size of those images to ~100 Kb each solved the problem.