Having browsed a few articles, the wobbly USB connector and failure to charge two PS4 controllers indicated a problem that may be difficult to repair. I wasn’t keen on hard-wiring the battery to the USB cable either!
Many suggestions seemed to conclude simply having to replace the controller in this case, until I found the link above with the simplest and best solution, a charging dock that uses the connector at the other side. Excellent. That should keep these good for some time yet.