This is just entirely false. As a counterpoint, I recommend this passage about the washing in the Texas Hill County. https://x.com/trainallday247/status/1796732464838160650
Sorry, we cooked with wood most of the time, kept animals, and even washed clothes and such by hand (no washing machines until much later). It really wasn't that big of a deal. And we weren't even that rural (next to decent sized city).
I'm sure they had it harder in Texas or Utah or Alaska. Not a universal thing.