I had the same problem with the terminology.
If distribution point is equivalent to cabinet, as in FTTC, then the amount of ventilation in the existing cabinets indicates a good deal more power is required than could reasonably be sucked down a few hundred yards of telephone copper and on the whole cabinets do seem to be located where power is readily available.
On that basis I assume they're thinking of something else. There's a BT manhole in the road opposite the end of my garden. My own phone line comes underground from that and the adjacent pole feeds about half a dozen neighbouring houses by overhead wires. Maybe that's what they mean by distribution point.
If the idea is to pull a single fibre to that they'll be pulling it through a duct which already contains a multi-pair cable which will be being made redundant so they could take as many pairs as necessary from that to feed power. Alternatively the fibre itself can be bundled with power lines.
And if they plan to run a bundle of fibres to that distribution point, one for each house then powering each house's box individually then power down the existing copper is probably feasible.