BT inherited the copper monopoly at the time, and legislation opened up their exchanges to competition. But you hit the nail on the head. BT innovate, run the infrastructure and handle all the external servicing. Even if you get your fibre product from someone other than BT, it's a BT engineer that comes out to fix it if something goes wrong.
BT do NOT have a fibre monopoly. Talking about them as if they did is reactionary. Virgin Media, while not available to 100% of the UK and saddled with crazy debt they're slowly paying off, are generally the ones to be beaten by BT, and not the other way around.