I think is not a novel idea: there are various vendors here in EU that offer "thermodynamic VMCs", meaning dual-flux ventilation system [1] with a small heat-pump between the to air flows, sucking heat from the outgoing air, heating the incoming flow in winter, in summer to avoid injecting too humid water they passively exchange outgoing fresh air with the incoming and the heat a water tank for sanitary water, cooling as needed the incoming air after the passive exchange, for instance
https://www.broferpura.it/en/products/Domestic/compact06.htm... this reduce the number of compressors a home need in mild climate and new homes.
For me I simply run the main heat pump at full power during the day on p.v. reducing the grid absorption for some hours in winter and for sanitary water almost zeroing it since 300l @60-65℃ (140-149℉) suffice for a whole day, at least when I have enough p.v. a day after another.
Honestly I doubt it's possible to store enough heat with current tech to compensate the home heating needs, it's possible for sanitary hot water (except for a swimming pool) but no more. However for mild climate for some months might be enough to significantly reduce the grid absorption. The sole issue here is that heat pumps now, without any specific reasons, have skyrocketed in retail price enough to make them far less convenient than just few years ago... So the economy is still not much.
[1] an image should be sufficient, I do not know their common name in English, sorry https://lacentrale-eco.com/images/blog/fonctionnement-vmc-do...