What is the Rosh Yeshiva’s position on showering on Yom Tov and bathing children on Yom Tov, both in terms of שווה לכל נפש and modern water heaters?
I shower on Yom Tov, it is clearly שווה לכל נפש and I don’t think גזירת מרחצאות applies. See an extended discussion in Shemirat Shabbat Kehilkhatah 14:7 footnote 21. I am not sure what your concern is regarding modern water heaters. Is it that it might be a grama for a new fire turning on? If that is the case, then as opposed to Shabbat, here it is only a derabanan of a creating a new fire, it’s a grama, and אינו מתכוין, so it would be totally fine. See the Shulchan Arukh OC 334:22 regarding gram kibuy on Shabbat when there is the potential for monitory loss. Mishnah Berurah extends this to all melachot. Poskim tend to limit that to cases where it is only occasional and also where the grama is not based on the normal way something operates. That could then be an issue here, if using the water heater could lead to a new fire igniting. That being said, as noted, it is only a derabanan and eino mitkaven (I’m not sure if it is pesik reisha or not, it depends on whether the new fire goes on every time). In general, most poskim rule that it is permitted to do grama for a new fire on Yom Tov. See Achiezer 3:60, Chatam Sofer EH 1:20, Yabia Omer 3:17, and Shemirat Shabbat Kehilkhatah 13:27, footnote 100.
As to bathing children, it is true that Rema in OC 511:2 only permits doing so if the water was heated before Yom Tov or in some otherwise permissible way on Yom Tov. However, given that I permit heating water on Yom Tov for a shower, since we shower every day, I would rule that if one bathes their child every day, that it is permissible to do so on Yom Tov. Most poskim don’t say this (just as they don’t address the shower issue), but Arukh HaShulkan (OC 511:5) does, and I would rule that way,
ויראה לי דזהו כשאין הכרח כל כך להתינוק אבל אם יש הכרח פשיטא שמותר בעצמו לחמם בשבילו דלעניין התינוק הוה שוה לכל נפש.