You won’t believe, because even I can’t believe that just 5 minutes after this post creation, (and just switching my own/admin’s interface theme to some other, and then reverting back), solved the problem.
I.e. wo any updating, or rebuilding, nothing, the site has started opening ok (full 24 hours passed during the problem). I don’t know if I should delete this topic or not (I’m anyway closing this).
Edit: The problem resurfaced next day. Sometimes there, sometimes not. Would report futher down.