Fire and air: cheap and bolt-up
These things are sold for a few hundred dollars and are intended to "bolt up" to any wall in the normal place a wall heater goes.
The "cheap" requirement forecloses any possibility of using efficient tech like a heat pump. They use simple (but rather expensive) resistive heating. That's wasteful.
The "bolt-up" requirement requires light weight and small size - so a storage reservoir of meaningful size is impossible - it would collapse the floor. Instead they use very high temperatures to store the heat in a sane-sized mass. The high temps won't play with heat pumps or passive solar, the only option at those temps is appallingly inefficient electric resistive heating.
That is your answer: they can't use water because they need to make the unit Very Hot to store a meaningful amount of heat in small space and mass.
Loss through insulation is proportional to
temperature differential / insulation value. So very high temps require very good insulation. Temperatures pushing 1000K preclude easy, cheap styrofoam or fiberglass, leaving the asbestos family of rock wools or other silicates. many of(and very balky insulation since most insulation does not like 1000K temps, leaving you in the asbestos family, yay).
You have to pay the piper, either in huge mass, or potent insulation to contain lots of heat. If you don't pay the piper, the device doesn't work very well.
Looking at the units I see on the Web, indeed, they do leak like a sieve, the convection units losing 80% of their heat via uncontrolled loss through the weak insulation. This makes them rather hard to control.
I remain skeptical that these small units can really time-shift heat the 8-12 hours needed to exploit evening electric rates. Where they show real potential is demand side management: the utility commanding heaters to turn off momentarily rather than spinning up a peaker. This ability greatly improves the resiliency of the grid, as it can effortlessly shed load if overloaded. If these are being pushed in your area, I suspect that is the agenda.
This for sure: because they store energy for such a short time, they are 100% energy efficient. Since there is no escape for heat but the living space, the only way these units fall below 100% is if they time-shift heat to times it is undesired. E.g. By leaking.
But they also cannot exceed 100%. We can do much better.
Water. Do everything, be everything.
Or safe propylene glycol antifreeze. "oh no, you can't run water at pyrophilic temperatures" Darn right. This is a completely different approach, and it is much cooler. The working temps would be 10-70C (in heat mode), so no need to cap the tank. Which goes somewher where leaks won't matter.
The delta-T being so small, we need a MUCH larger thermal mass but insulation losses will be 1/10. We also benefit from square-cube law: cube the volume, only square the envelope size.
You install a large tank, either as a cistern (with proper drainage) or outdoors and make up for it with better insulation. Any common agricultural tank will do. 12"+ of styrofoam would be good. The delta-T is 60C not 600C, so the insulation could be less, but it's easy with materials like styrofoam available, so go for a lot more. You'd use a hefty sized tank right-sized for your load - 1000 gal. would not be excessive if your design engineering called for it.
You have many options to add heat to such a low-temp tank - fuel, solar collectors, resistive electric if you really want to, but the darling will be heat pumping because it will run 200-400% efficiency. In fall/spring, the heat it pumps in at night could be the heat you rejected by day using its air conditioning mode.
Then you'd use heat pumps to transfer the heat into the house - if the tank is 60C and you want your heat pump outputting 40C air, that heat transfer will be VERY efficient because it is "downhill" for the heat pump. You're simply using the heat pump to transport the heat from place X to place Y.
Heat pumps which do this aren't uncommon, they are used in large complexes which circulate "service water" at 70 degrees for both A/C and heat. In summer they use cooling towers, in winter the boiler room heats the water with fuel (gas).
I mentioned "heat mode". Heat pumps reverse into air conditioners, so this system comes with A/C. At night your heat pump chills the tank, dumping to ambient or house heat. If the tank can abide freezing, water wins massively in A/C mode: the enthalpy of fusion lets you store an insane amount of cooling power. (too bad you can't find a substance that freezes at 25C). By day, you pump heat from the house out to the tank - "downhill" again. If dump freon temperature exceeds ambient outside air, you can cool to ambient before dumping in the tank.
Earth: build the thermal storage into the house
In this case, you use an even lower delta-T and even more storage material: the masonry construction materials of the house itself. This is obviously not a bolt-on, but passive thermal design from the outset.
In this case a high-thermal-mass construction material is used, and generous insulation is placed on the outside. As a result the building envelope strongly resists change to interior temperature, and itself stores the heat that we are time-shifting.
Dealing with only a 5-10 degree delta-T, proportionately more mass is needed. Insulation has less work to do, but it is still important.
Implementation is easy: any commercial off-the-shelf HVAC equipment will suffice. Using the thermal storage is just a matter of timing thermostat settings.
If that does not suffice and the homeowner wants something more on-demand, it would combine nicely with a water system, and the "water" could even be an enormous chunk of concrete.