The cold water getting cooked indirectly is also “Lo neecha lo”
In addition to turning off the themostat, shut off the cold water intake of the water boiler tank to resolve the bishul issue.
Rav Ruderman zt’l, the Rosh yeshiva of Ner Yisroel did not generally paskin but would refer shailos to Reb Dovid Kronglass zt’l.
truthsharer1 – “I have heard of people leaving a faucet running with a tiny trickle of hot water to allow normal usage on Shabbos, but I don’t like that loophole, you are still causing more cold water to heat up as soon as you turn on your faucets.”
Rav Nachman Bulman zt’l had the hot water going over Shabbos as per your post.