Making a pikuach nefesh phone call on Yom Kippur - mitzva or something to be avoided?

Question

Outside of our synagogue on Yom Kippur, someone fainted and a doctor present decided to call an ambulance. At that time, there were multiple people present and one phone.

Should people have jumped to "grab the mitzvah" and be the one to call emergency services? Or would it have been better for the individuals involved to be passive and wait for someone else to make the call?

In other words, is making such a call

• a mitzvah to grab?
• something to do only if that is the only solution?
• something to be avoided unless really needed?
• something else?

Remember to please provide sources.

Answer 1

מְפַקְּחִין פִּקּוּחַ נֶפֶשׁ בְּשַׁבָּת וְאֵין צָרִיךְ לִטּל רְשׁוּת מִבֵּית דִּין. וְהַמַּקְדִּים לְהַצִּיל הַנֶּפֶשׁ הֲרֵי זֶה מְשֻׁבָּח.

[All] activities necessary to save a life should be performed on the Sabbath; there is no necessity to receive a license from the court. The more zealous one is [in this regard], the more praiseworthy. (2.16)

כשעושים דברים האלו אין עושין אותן לא ע"י נכרים ולא ע"י קטנים ולא ע"י עבדים ולא ע"י נשים כדי שלא תהא שבת קלה בעיניהם. אלא על ידי גדולי ישראל וחכמיהם. ואסור להתמהמה בחילול שבת לחולה שיש בו סכנה

When such treatment is administered, it should not be administered by gentiles, by children, by servants, or by women, so that they will not view the Sabbath flippantly. Instead, the treatment should be administered by the leaders of Israel and the wise. It is forbidden to hesitate before transgressing the Sabbath [laws] on behalf of a person who is dangerously ill. (2.3)

Rambam Shabbat 2,16 (on Sefaria but Chabad have a better translation here)

Rambam continues that this is done even in doubt.