BT DO still have the monopoly on the final mile to your house - that's what LLU is all about.
BT still need to connect (or reconnect) this final wire, which clearly existed at some point in the past.
The PO (which is completely unrelated to BT these days) are just the same as TalkTalk or any other phone provider.
The point at issue is what BT will want to charge for reconnecting the line into the local exchange - and whether it will be waived if you promise to use them as your phone service provider to boot, for a minimum period of time.
If the PO are making this offer, they must be doing it by giving you �10 back off the BT reconnection price - its the only way - because BT alone control this final mile.