This answer asserts that R' Auerbach held that not using electricity on Shabbat is a minhag. However, assuming that this ruling dates to the 1930s (which is what I have heard in the past), everyday usage of electricity went back only about 50 years.
Based on this information, how could R' Auerbach assert that not using electricity on Shabbat is a minhag, considering its [electricity's] relative novelty?
Journal of Halacha & Contemporary Society, No. XXI - Spring 91 - Pesach 5751 has an article by Rabbi Michael Broyde - Adjunct Assistant Professor of Law at Brooklyn Law School, and Rabbi Howard Jachter - Associate Rabbi of Congregation Beth Judah in Brooklyn which gives Rabbi Auerbach's opinion in his own words. The article concludes that current practice is that it is forbidden in any case.
The original quote from Rav Auerbach (Minchat Shlomo 74, 84) is
In my opinion there is no prohibition [to use electricity] on Shabbat or Yom Tov... There is no prohibition of ma'keh bepatish or molid... (However, I [Rabbi Auerbach] am afraid that the masses will err and turn on incandescent lights on Shabbat, and thus I do not permit electricity absent great need...) ... This matter requires further analysis.
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However, the key point in my opinion is that there is no prohibition to use electricity on Shabbat unless the electricity causes a prohibited act like cooking or starting a flame.
The article goes through the various opinions on using electricity and concludes, in the relevant part:
Rabbi Auerbach additionally states that since the tradition forbids the use of electricity, and this tradition received near unanimous approval from rabbinic authorities in the normal course of events observant Jews should accept this tradition (even though he feels it is based on incorrect premises) and operate under the presumption that the use of electricity without light or heat is a violation, of rabbinic origin, based on molid.
