I've heard it said that the majority of American women didn't want the right to vote because of legitimate reasons such as not wanting to ever be conscripted during war. Is this true?
By most reasonable framings of the question the answer is "probably not". (It's much more difficult to say what the answer would have been before women's suffrage became a political issue.)
It's hard to get polling data from that early in the 20th century, particularly of women. But we know from the passage of the amendment male politicians supported it, and we know from suffrage referendums in various states (the only one I can find the exact numbers for is Colorado, but I know California and New York also had statewide women's suffrage by referendum by the time of the 19th amendment) that a majority of ordinary men also supported suffrage. Since most of the protesters for suffrage were women, most of the major activists for suffrage were women, and the average supporter of suffrage at the time was stereotyped as a woman, it would be very strange if a majority of women didn't support it despite women apparently being much more fervent supporters of suffrage.
As democracy arrived early but female suffrage ultra-late in Switzerland (only in 1971 at the federal level thanks to a majority vote by the male population), it can act as a sort of time capsule for this.
Surveys in Geneva (1952), the city of Basel (1954) and the city of Zurich (1955) showed that clear majorities of women were in favour of suffrage and the argument that women did not actually want the right was wrong (that said, these polls were taken in progressive and urban areas, which then introduced female suffrage on a local and state level in the 1960s).
In 1958, a Swiss women's committee against female suffrage was constituted ("Schweizerische Frauenkomitee gegen das Frauenstimmrecht", "Bund der Schweizerinnen gegen das Frauenstimmrecht") to fight against their own right to vote.
German language newspaper article about 40 years female suffrage in Switzerland (with anti female suffrage posters) http://www.nzz.ch/frauenstimmrecht-1.9350588
Official documentation by the Swiss federal commission on women's issues in German, French and Italian http://www.ekf.admin.ch/dokumentation/00444/00517/index.html