The source which told you that should also have explained that you can actually have women voting in direct democracy as any other citizens, there is nothing preventing that. You don't need to do it in person on a square, either.
(It's a known fact, yes, in general, people do vote more socially conservatively in referendums, often backing up status quo. But that doesn't prevent progressive politicians to come up with better proposals.)
In any case, if we use your logic, US would be perfect country for this, being one of the last countries on Earth that doesn't have universal health care system.
Interestingly, many U.S. states do have some direct democracy provisions, courtesy of the progressive movement at the beginning of the 20th century. But what I heard it was sabotaged at federal level by the administration at the time, because U.S. wanted to get a bit involved in WW1 and it could potentially prevent that.
The place where women couldn't vote in federal elections until 1971, and in local elections as late as 1990?