How do all of those factors compare with say the sodium ion or sodium sulfur battery?

This is the concept behind combined-cycle generation, where an initial stage (direct natural gas combustion) runs a turbine whose cooling (water) drives a second-stage steam turbine. These can push total efficiency to ~60+%, with an ultimate efficiency of ~90+% possible where a tertiary thermal application which uses low-grade heat (anywhere from ~100--180°C / 212--356°F) can be used for some industrial, food-process, or space-heating applications, called cogeneration.

<https://en.wikipedia.org/wiki/Combined_cycle_power_plant>

<https://en.wikipedia.org/wiki/Cogeneration>

Seimans has a good video 'splainer: <https://yewtu.be/watch?v=eeiu-wcyEbs>

GE also has numerous offerings in this space, and I've seen trade articles describing this in the past, though I'm not finding any presently.

There isn't some majyckal process by which all thermal energy can be converted to motion (or by extension, electrical generation), but it is possible, with additional complexity and capital equipment, to extract much of it.

In winter I think the plan is to use the heat for district heating.

(which is also kinda interesting since in parts of Switzerland district heating through waste management facilities starts to be a problem due to a reduction of the amount of waste available)