That mostly did the trick. By mostly, I mean the controller get initialized in the main menu, but the D-Pad of the controller doesn't do anything after that. The rest of the buttons seem to work fine though, although using the left analogue stick for directional buttons is not in my opinion the optimal experience while using a gamepad.

I don't find any info regarding using the J key in the main menu to check for controllers again in the JJ2+ Readme file online though. Could this be added to the section "Features and fixes" maybe?

While talking about the section "Features and fixes", if you ask me, other noteworthy inclusions there could be that the D-Pad of the Xbox controller now can be used with Jazz2+, as well as the Back button on the Xbox controller now acts as Escape. I believe that info can be found under "Latest Changes" somewhere, but it takes quite some time to discover those additions in the changelogs on the JJ2+ Readme page.

I remember before those things were build-in features of Jazz2+, that I had to use other applications to map my Xbox controller while playing this game to get those exact functions that Jazz2+ now provides.