I did a fresh install and started installing the mods one by one, testing after each one.
This allowed me to narrow the problem down to an interaction with EFT API (which was installed as a dependency for Game Panel HUD, though the problem is reproducible only with EFT API installed and no HUD mod.)
If either EFT API is not installed or the stance setting in realism is disabled, things work fine, otherwise the problem occurs as described in my post above.