Or you just do what happens now, and use that power to boil kettles, run TVs, run factories etc.
Energy storage has a place, but there's no reason it needs to be directly connected to railway catenary.
On your second point, 25kV isn't the best choice for long distance power balancing, and it would be complicated by the fact there will still be some trains running, with a very peaky power demand. I also don't think the ECML, for example is a single circuit all the way from London to Edinburgh, it's a lot of much smaller sections separated by neutral sections.
It's designed so that feeder station feeds a single section, so by definition it's useless for moving power around the country.
I'm puzzled as to why we need some sort of not very good duplicate for the National Grid?
The logical place to locate batteries or other energy storage is near to very large renewable. It can already be moved wherever it's required by the grid, railway electrification schemes are designed for powering trains, not duplicating the grid.