Turns out the Mac is connecting to the desktop via Bluetooth 3 not 4. So the problem is, the Mac Mini is not supporting Bluetooth 4.0
This is a 2013 unit and 4.0 was supposed to be implemented in 2011. The usual test for 4.0 support (checking LMP version is 6) indicates it does support 4, but it simply does not.
It seems Apple have been sending out incorrect info on this. Although this is now out of warranty I might start to make some noises in the direction of the official Apple dealer I bought it from.
Is Bluetooth 4 compatible with Bluetooth LE? Afaik is the BLE a Bluetooth 3 version, but different specs.