Okay, I’ve pinpointed the issue to the theme I’m running. The reason it may be causing issues is that it is not really a normal theme, more like a collection of a bunch of plugins (also that’s why it’s only available on GitHub).
Do you know what I should be looking for that could be causing the issue? This theme does a bunch of stuff with having built-in permissions manager to playing around with page templates and custom post types so I cannot really just stop using this theme.