Why did Switzerland not grant votes to women until 1971?

Well the "Reason" to this was the Swiss vote system. To make a change in the Swiss constitution, a "vote initiative" has to be submitted. If the prospective vote initiative fulfills some conditions and other things, it gets to be an initiative. This initiative goes out and then the people which are allow can vote about it. They can accept or decline it. If they accept the changes will be made, if not, they won't be made.

Until 7 Feb. 1971, only Swiss men could vote about such an initiative. So simply the Swiss men were the reason why women couldn't vote in Switzerland until 1971.

Before the vote in 1971 it was likely that the vote could end in a NO, so the opponent's of this initiative tried to stop it. Very successfully as we know today. But in 1971 it was more likely that the initiative would be accepted, so the opponent's stepped back from fighting against it (so that they wouldn't lose potential voters).

Due to this, the initiative was accepted more easily, but please be aware this was for the "country" level. The catons Appenzell Ausserrhoden, Appenzell Innerrhoden, Glarus, Obwalden, Schwyz, St. Gallen, Thurgau, Uri still declined to grant woman the right to vote in their Cantons.

If you can read German/Italian/French there is a official overview about the fight for woman rights in Switzerland here: https://www.ch.ch/de/wahlen2015/zum-50-mal/warum-konnten-die-frauen-in-der-schweiz-erst-ab-1971-abstimm/

In Switzerland, it had to pass a popular referendum. (Switzerland also joined the UN and legalized abortion only in 2002 — both decisions that were passed through referendum.) Similarly in Liechtenstein, 1984, where the 4th referendum in 16 years only narrowly passed despite support from newspapers and both major political parties.

Passing a popular referendum is a much higher bar than say passing a law (e.g. UK, 1918) or a constitutional amendment (e.g. US, 1920). (Edit: I don't mean in general. I mean just for the issue of women's suffrage. Just my opinion.)

And in France for example, the Committee for National Liberation simply gave women the right to vote in 1944. Were it by referendum, it would probably have taken years more before French women had the right to vote (though perhaps not as late as 1971).

One might then ask, "Why are (male) legislators more inclined than the general male population to give women the vote?"

Thinking selfishly and ignoring the moral element, if you are a man on the street, giving women the right to vote simply means your voting power is halved. This is purely a bad thing.

But if you are a legislator, it is unclear whether women voting is necessarily a bad thing. Indeed with women voting, this might be a good thing for some legislators, who can now get a bigger proportion of votes.

Edit in response to Mark C. Wallace:

I'm looking for an answer on why 1971 succeeded - why didn't it happen in 1965 or 1995. What happened that made the 1971 attempt successful?

I think it was less "something special or dramatic happened circa 1971" than "gradual evolution of social attitudes" (not unlike what we see in various countries today with gay marriage).

Here's the timeline suggesting just such a gradual evolution:

1. 1959 national referendum: Rejected 66.9% to 33.1%.
2. Introduction of Women's Suffrage at the Cantonal Level. 1959: Vaud and Neuchâtel. 1960: Geneva. 1966: Basel-Stadt. 1968: Basel-Landschaft. 1969: Ticino. 1970: Valais, Lucerne, Zurich. 1971: Aargau, Fribourg, Schaffhausen and Zug.
3. Feb 1971: National referendum passes 65.7% - 34.3%.
4. The Appenzells only gave women the right to vote (cantonal) in 1989 and 1990.

I think asking why the Swiss referendum succeeded specifically in 1971 is a bit like asking why regarding gay marriage in the US, Obergefell v. Hodges "succeeded" specifically in 2015. Or why there was a crossover point in support for gay marriage in 2011: