Oh nevermind - I managed to get it to work!
The thing is, there are two ways to import from other Browsers.
The first one is Settings > General > Import Browser data
. That's the one I tried to use, and it doesn't work no matter which browser or which data, as explained above.
However, while looking around the UI I found a second way: Menubar > File > Import From > [Browser]
. When I import data via this route, it works with all browsers and all data!
@MarkNashed try that, maybe it works for you as well! and @Vlad maybe this helps you to reproduce the issue - there seems to be some differences between the two import methods?