My initial thought about this was that it's using fuel to make electricity, right? Rather than using sunlight/hydro/etc -- kinda like a generator, but without the mechanical aspect?

To my limited understanding yes, that's what they claim.

Basically burning fuel (any fuel, really) with added sodium to create very bright monochromatic light that can then be converted into electricity using very high efficiency solar cells.

It's a different way to capture energy stored in fuel.

Normally to produce electricity from fuel you would spin a turbine, either with a mechanical engine or using vapour. But here the energy is captured through a photo cell, and the author claims that mixing sodium into certain fuels leads to a very significant part of fuel energy going into light at specific wavelength.

From the "wire-to-wire" mention it seems that they're proposing using electric power to generate and store hydrogen till it's needed, then burn it to get electric power back.

But they say other fuels work, in which case it wouldn't be "wire-to-wire", and then it'd be more appropriate to compare this to a power generator fueled by natural gas or gasoline. A generator with no pistons or turbines, just a fuel pump, sounds fantastic, if they can make it work. But you'd have to supply sodium.