I'm very much rooting for this. A lot of train tracks are not electrified simply because it is not economical to do so. I know that most of Denmark(outside of greater Copenhagen, and Aarhus) is still operating Diesel trains. Such battery backed "electric" trains may be just the thing to turn those Diesel trains into cleaner ones without needing a huge capital investment of electrifying vast track networks that are not used justifiably as much.

New battery powered trains unfortunately are even more expensive of a capital investment. Especially when it comes to maintenance over classic electric trains.

What I always found weird is that a power rail system isn't used. One could do it even without a third rail...

How can it be done without a 3rd rail?

I can imagine running electric wires along tha tracks is a big challenge. However I think using the container BESS mentioned in the video above receiving trickle charge from the grid would leave opportunities for local solar generation to augment the grid too. Think of it like a decentralized solution. Scales much more and provides resilience against outages.

Wiring up the tracks shouldn't be expensive. See there's two metal tracks, just use one for the positive line and one for the negative line. ;)

Well, the tracks are normally gapped, they still need to be adapted to be high current continuous.

(strictly technically speaking not direct to grid but yes)

The problem is gaps in electrification. So you might have electrified two parts, but with a big gap without wires

This lets the train cross those (and mind you this would have to be something a larger gap, if it's a smaller gap without stops, let's say 1km, then the train just doesn't care)

Normal electric trains already carry batteries for this reason, and to be able to manuever off the track in case of a power outage. Much like trams.

There's a big difference however between a train with a short term battery backup and one that rides on pure battery power for an extended range.

It significantly increases the cost of track per mile. The (then) Soviets started electrifying their railways in 1930, and didn't finish until 2002.