Can’t you allow the use of hot water from a faucet by arguing that it’s a case of davar she’anyon niskaven and p’sik reisha d’lo nicha lay ("d’lo ICHPAS lay), and therefore we don’t care about cold water that will get heated up if we’re only using a relatively small amount from the boiler?
Shalom, Thank you for your question. The issues you raise concerning use of hot water on Shabbat center around some of the most basic halachic principles, and as such have been widely discussed by many rabbis. Probably the most in-depth examination of the issues can be found in Rabbi Ovadya Yosef's Yabi'a Omer, volume 4, Orach Chaim, responsa 34 and 35. Even though one could say that when turning on the hot tap the cold water that enters the boiler as a result is only a pesik reisha that is "lo nicha lay" (i.e. an inevitable event that he does not care about), nonetheless the overwhelming consensus of rabbinic opinion is to forbid this. This is because we are dealing here with a Torah prohibition. The cold water that enters the boiler will be cooked by the hot water already there. This is considered cooking from the Torah. So, even if this were done by way of a pesik reisha that is lo nicha lay, we rule that it is still forbidden (at least rabbinically) [unlike those rishonim who allow a pesik reisha de'lo nicha lay, the Shulchan Aruch rules that this is forbidden - see Shulchan Aruch, Orach Chaim, 320:18]. See Shmirat Shabbat KeHilchata (Chapter 1:39), and Yalkut Yosef, Shabbat, 318:31 who both rule that it is forbidden to turn on the hot tap with such boilers. If the water in the boiler was heated up via a solar heating system (what is called a "dud shemesh" in Israel) then, as cooking via water heated up in the sun is only rabbinic, the arguments you quoted are more readily applied. In such a case there are many rabbis who do rule leniently and allow the use of such water (though some forbid it for other reasons). Blessings.