Because it's not true. Maybe mid-1800s, but by mid-1900s women could totally use cash. Not credit cards though, 1974 is when women were allowed to get those. In other words, the exact opposite of what that comment is saying.
Up until the invention of the self-service supermarket you could not walk into a grocery store, pick out a can of Campbell's Chicken Noodle Soup, flip a dime to a cashier, and walk out with it.
The self-service grocery store did not exist until WWI.
They did not become common until some time around WWII.
They did not go extinct until the 70s.
My grandmothers could not (and did not) use cash to buy groceries because there were no cash-and-carrys near them. They needed an account, until the 60s-70s in rural Tennessee and Missouri where they lived.
Everything was paid by check. Drop off the order, pick up the order, bring home the bill, the husband sends in a check.
There was an exact and precise 0.0% chance in many parts of the United States of America of a single unwed woman getting an account at a grocery store without a male cosigner.