Many thanks to all those who responded, first of all.
Based on the initial comment by @wally, I made some further tests yesterday night. Indeed, the car kit has a button/function to import the contacts from the phone, but it fails to import anything. Its message is that maybe I need to authorize the sharing on the phone.
I also tried to export all contacts to vCard on the phone (successfully), then to browse to that file in file manager, then shift-left and click on the sharing icon, but that only listed "Messages" and "Dekko" as potential targets. New import attempts from the car kit ended up with the same error message.
So I guess that this "sharing" icon is where a "BT File Transfer" app would need to register itself. For the record, this is how I used to send contact info to the car kit from my Nokia N900, using a BT transfer app called "Petrovich'.