> How about invent a hot water system that can be
> used on Shabbos. We have urns for coffee. Why not
> make the actual boiler work in a similar way so that
> you can turn on the hot water in the sink/shower to
> wash dishes etc.
Yes, this does already exist. I lived on a Kibbutz in 1976 where they designed/installed such a device on a hot water heater. When in switched to “Shabbos mode”, it kept the water temperature in the tank below yad-soledes.
However, I don’t know whether such a device would help regarding the issue of showering on Shabbos/Yom Tov. That may depend more on whether poskim consider this change sufficient to relax the original gzeira.