Heavy electricity lines are invariably in built up areas, not in open countryside. Not erecting OLE on the Chiltern will save a fortune. The DfT did state a few years ago they will not be extensively installing OLE looking into other forms such as battery-electric hybrids, etc.Stations are not necessarily near any 'hefty' electric cables. A simple station can be fed from the grid as a domestic supply. A busier station may have a bigger connection but still doubtful it'd be of any use for electric charging (unless you went through the 'trickle-charge a shore reservoir and dump' option, which has it's own downsides)
You also need time to charge.
For third rail, yes. I don't see any 25KV AC OLE being taken down, primarily because we don't really have enough of it as is. I can, however, see lines that should be electrified being skipped because of batteries (frustratingly).
Merseyrail has around seventy miles of 3rd rail. I can all that being removed for battery trains. Apart from the rail tunnel under the Mersey which has very steep inclines which OLE can be used. Trains can top up the batteries at stations as they go along if need be, using OLE technology.
Manufacturers state that their trains can reach high speeds on batteries with long ranges, such that the Chiltern can use battery hybrids. Battery hybrids could be used on the London to Chester route on HS2 track. The thirty miles of line from Crewe to Chester is unelectrified, where batteries could take over.