The Rashba says the reason a man is saved first is based on the pasuk in Vayikra 25:36, where the pasuk tells us the “brother shall live” with you. The Rambam says the reason is because men have more mitzvos so they have a more important function in life, as a man is more “mekudash” than a woman which gives him precedence להחיות. The Shach (Y.D. 251:11) explains that the definition of להחיות is to save them from a life threatening situation. Also see Rama YD 252:8 and Taz YD 252:6 saying the same precedence is applicable.
The Ya’avetz (quoted in Pischei Teshuva YD 252:7) asks what would be if the threat is not physical but rather spiritual and the community has a choice to redeem either a boy or a girl, but not both, from the spiritual danger. The Ya’avetz concludes that since the precedence is given to the boy for physical danger, it should certainly be given to him for spiritual danger.
The Igros Moshe 7 CM 75 also rules this halacha is applicable. In fact, there’s virtually no disagreement whatsoever among the poskim that this is a valid and applicable halacha. Rav Moshe writes that “that it should be obvious and clear to every Ben Torah and Yirei Shamayim that one is obligated to cure and save every person without regard to their wisdom or knowledge and one should only triage on the basis of the order in the Mishna in Horayos”. (Igros Moshe CM2 74:1). Rav Chaim Rapoport in “The Halachic Hierarchy for Triage” (Le’eyla, June 2001, pp. 27-38) writes “Rabbi Feinstein’s statement clearly reconfirms the unchallenged halachic system: the criteria in the Mishna in Horayos remains binding in contemporary times and have not been displaced by any ‘custom’.” The reluctance by some to use this system, Rav Rapoport says, is an aversion to this politically incorrect and non-egalitarian halachic ideology.