History does not support their position.
Historically, men did not have the universal suffrage that suffragettes demanded for women. Voting rights were tied to all kinds of terms and conditions. Further, there are examples (with more emerging) of women voting before the women's suffrage movement.
One such example is a recently discovered document listing English women voters in an election which took place in 1843, 75 years prior to legislation recognizing women's voting rights in 1918. At that time, suffrage for men was not universal, but limited to the upper classes, with various groups agitating for parliamentary reform throughout the 19th and early 20th centuries. The women recorded in the 1843 document would have had to meet the standards met by men. They paid a fee, and it determined how their vote was counted. Note that the article mentions that the high fee paid by Grace Brown gave her 4 votes, where those who paid less got only one vote. These women also enjoyed a privilege denied to men who did not meet legal voting requirements. Adult men who were not heads of households could not vote.
Prior to the formation of the United States, voting in the colonies was largely governed by the same standards used in England. However, contrary to popular belief, women were not universally barred from voting. As with the women in England's 1843 document, American women who voted prior to the 20th century did so under the same terms and conditions faced by men, save for one: Women were not and still are not subject to being drafted into the military in times of war.
One interesting example of early female voters prior to universal male suffrage is the colony of New Jersey, where gender was not a factor in voting rights until the Democratic-Republican party, which eventually became the Democrat party, took the vote away from Jersey women, minorities, non-citizens, and the poor in 1807 over conflict between their party and federalists.
Even after the U.S. became a nation, men's suffrage was not universal. Voting rights continued to evolve throughout the 19th century, with states slowly letting go of property ownership requirements over the course of decades. After the 15th amendment was ratified, recognizing black male citizenship and voting rights, southern states passed "grandfather clauses" to roll back their rights, and used Jim Crow laws and poll taxes to get out of recognizing them until the success of the civil rights movement in the mid 20th century. This allowed wealthy and middle class white women to vote while many poor and minority men and women were kept away from the polls.
In 1876, the supreme court ruled that Native Americans were not citizens as defined by the 14th Amendment, and therefore could not vote. In 1890, they were told they could apply to become naturalized citizens in their own ancestral land. Laws denying citizenship to various Asian immigrants passed in 1882 (the Chinese exception) and 1922 (Japanese immigrants.) In 1919 Native Americans and in 1925 Filipinos were told they could earn citizenship by risking their lives serving in American wars. Various stipulations, including Jim Crow laws and poll taxes, left the majority of the indigenous population of the U.S. and its territories, along with the majority of Asian immigrants, and most minorities, subject to the rule of the American government without representation by officials for (or against) whom they had the right to vote - the same injustice that sparked the Revolutionary war. Asians did not see their voting rights universally recognized until the McCarran-Walter Act of 1952. Native Americans' right to vote was not fully recognized until 1957, 37 years after the 19th Amendment recognized women's right to vote... 3 years before the first man walked on the moon. It was not until 8 years after Native Americans were recognized, 5 years after we put a man on the moon, that the Voting Rights Act passed, protecting the right of blacks and other minorities to vote. Upper class white women got the vote in 1920. Impoverished black men and women did not truly get theirs until1965.
Feminists who paint the history of citizen suffrage in starkly gendered terms do so in either ignorance of or contempt for reality. Western civilization's growing pains have not been so clearly defined, nor has the weight of feminism's influence on the evolution of voting rights. It was both separate and tandem fighting by many disenfranchised groups which brought about voting reform in England and the United States, including, but not limited to, women's rights.
The suffragette movement is yet another example of feminists approaching a human problem as if only women are impacted, and only women deserve relief.