Unplug the broadband connection to the home hub, then ring your phone number from a mobile phone and see if you still get engaged tone. Make sure your phone is still plugged into the socket on the wall.
Its possible you have a short circuit on your phone wiring which is why you would get engaged tone when you call your home number.
Its possible for the home hub to cause a short circuit, which is why I suggest disconnecting it.
Next thing to try is to unclip the front cover on the NTE5C, and plug your phone directly into the socket that is showing as that will eliminate any issue with the NTE or any extension wiring.
There is no other way for the Openreach Field Technician to test, apart from the test socket unless they disconnected the wires in the junction box and tested from there.
There is also a possibility of an intermittent fault in the exchange, as the equipment is getting very old. This could also cause some of the issues you are reporting.
Just in case you are not able to progress this, I have asked a moderator to offer help.
