I'm unaware of a restriction stopping another company running their wires locally if they wish, but no one seriously expects them to do so. They be forever, if at all, recouping the cost. However cable firms have run fibre around the country is you are looking for an alternative.
Yes the situation is not good when the network owner is different from the service supplier because they can point at each other. And since BT lost the real monopoly and had to let others use their network they are less keen on doing more than remote checking without warning you that if it proves not to be a BT fault you are responsible for all the costs. Thus is one of the costs of having competition.