That information would have been useful a few pages ago.....
Despite the scooter being and S3 version I suspect they use the same controller throughout the range therefore the schematic for the SE is likely to be the same: assuming this IS the case.........
View attachment 38360
You can see that the charger socket (CH1) has the three pins, pin 1 negative, 2 is positive and 3 is the charger control line. This is the yellow/green wire from the back of the socket shown in earlier pictures you posted.
Since you state:
then your charger is obviously ok.
The 'missing' item is the yellow/green (charging) signal wire which, I believe should be shorted to the negative wire.
With the charger OFF but plugged in to the XLR socket, measure the resistance between pins 1 and 3 (meter on ohms/continuity) and check that there is a short between them.
If there is then there is a fault on the 'rear board' - chase the yellow/green wire to JP3 to make sure it is connected and also check F3(? can't read it on the diagram but its labelled as 'lights fuse') as, strangely, this may be part of the charging circuit?????? < probably not, but no harm in trying.
Either way, the lack of voltage on your charger output is perplexing - the charger doesn't lead me to believe it auto-disconnects its output without a load connected but this 'may' be the case and therefore leading us to believe its faulty when measured disconnected but, in fact, it's not.