Same problem here. IOS 17.3.1 now and still no fix. Ford Kuga, UK, Sync 3, software version 2.3. The phone sometimes connects, sometimes doesn't. Sometimes rapidly connects and disconnects as you described. Tried multiple cables, USB C-C with adapter, C-A, low spec, high spec cables, all the same issue.
To try and work out whether it is hardware of software related, I tried to connect my old iPhone 13 Pro (now my daughters), Lightning to USB-A running 17.2.1 - same issue. So it does seem to be an iOS 17.x.x issue and not a hardware / USB-C specific issue.
Any response from Apple on this???