Pikuach Nefesh on Yom Kippur

Hello Rabbi,
I heard a recorded lesson by Rabbi Zilberstein Shlita, in which he told of an elderly doctor who was called in the midst of Yom Kippur to give a sedative injection to a man who had suffered a herniated disc and was suffering from severe pain. The doctor hesitated whether to go, because it was on the other side of the town where he lived, and walking there and back in the intense heat, plus his advanced age, which was over 70, would certainly cause him to break the fast, eat or at least drink. And this case is simply not a case of pikuach nefesh, but only severe pain that is not life-threatening.
Rabbi Zilberstein said that he approached Rabbi Elyashiv ztzu”l with this question, and Rabbi Elyashiv ruled that the doctor did indeed have to go and inject the patient, from the perspective of “You shall not stand on the blood of your neighbor,” and from the perspective of “and you shall return it to him,” since the loss of his body is greater than his wealth, and since this is what he is commanded to do now, he does not need to take into account what he will have to do later to break the fast, this fast on his own. He emphasized that a person is commanded to do what is in front of him at the moment!
They asked him there in class what about the famous question, about a person who cannot fast both the Gedaliah fast and the Yom Kippur fast, according to the halakhah it is ruled that he should forgo Gedaliah so that he can fast on Yom Kippur. And apparently why was it not said, as here, that he should do what he is commanded to do at the moment, and right now the Gedaliah fast is before him, and what he will later have to forgo on Yom Kippur is already a calculation in itself.
I didn’t really understand his answer there, and I would appreciate the Rabbi’s answer to the division between the cases or the underlying reason for the first case if the Rabbi agrees with this ruling.
Thank you and good signing!