As your line works perfectly on the test socket, then the fault is with the internal wiring as other have suggested
Disconnect all of the extensions and only use the one socket, get wireless phones for other locations in the house and plug the base station into the main socket along with the router
Replace all of your house extension sockets with better cabling, if the extensions were made say 10 years ago then the cable is not suitable for broadband use and is interfering, if it has been done recently then it’s of poor quality and you should complain to whoever installed them
I have dealt with this problem lots of times, I generally just go for the cheap option, in one case I got them 4 DECT phones for £60 and disconnected all their extensions. I just placed a dect phone in each of the locations that had an extension socket; speed went from an unstable 1mbps to a stable 4mbps
Thank you for all the advice.
A little more fault-finding identified a problem with my Calor tank level sender that attaches to the line. I removed that completely and all is well. I haven't checked the extension yet but the silent-line test seems okay.
I'd like to use the hub on the extension as we get much better wireless coverage through the house so I will try that later.
In theory there is no reason why you cannot use the hub on an extension so long as EVERY phone socket has a filter (if the socket is used).
However ADSL quality is related to line length (as well as line quality) and so you will reduce your maximum speed probably (depends) so you need to watch for that.
Personally what I do is have my router as the master socket but have another router elsewhere that is set to DHCP relay mode and used to control the wireless network only, that way I can have it where best located.