The tosafos on amud b says there is no isur of mevashel when pre-heating asasiyos shel barzel and throwing them into the kohen gadol's mikveh on yom kippur (Yoma 34b). Why is that? My rough calculation is that in order to raise the temperature of 600 litres of water from 15 to 16 degrees centigrade we need a 2-litre iron piece of 375 degrees. Obviously, the asasiyos were very hot and could instantly and simultaneously boil a volume of water more than kedei lirchotz ever katan. One could say that this is a davar she-eino miskayem because the hot water immediately dissolved in the rest of the mikveh, but I don't see how it's relevant here, obviously we are interested that this boiling water raise the average temperature.
Thank you
Akiva, London
1) Heating water to 15-16 degrees centigrade is not a problem. The Gemara says here (40b) that one may place a flask of water near the bonfire, not to heat it up, but just to take off the chill. The Gemara there (40b) also tells us that Rav Yehudah said in the name of Shmuel that if it is less than the temperature of "Yad Soledes Bo" (the "hand pulls away"), this is permitted. Shemiras Shabbos k'Hilchoso (chapter 1, note 3, in the old edtion) cites Rav Shlomo Zalman Auerbach zt'l who says that below 45 degrees centigrade is certainly less than Yad Soledes.
2) Even though a small amount of water near the Ashashiyos can reach Yad Soledes, this does not disturb us. The Tosfos that you have cited, Akiva, writes that the heater cannot cook most of the water in it. We follow the majority of the water. Tosfos agrees that a minority of the water could be cooked, but this does not present a difficulty. The reason is, as you write, that the cooking on this small quantity does not have any Kiyum. The cooked water mixes immediately with the rest of the water and cools down. The Shulchan Aruch (OC 314:10) writes that if the Melachah does not last at all, this is permitted l'Chatchilah.
Akiva, wishing you a healthy summer!
Dovid Bloom