To launder and doing the laundry mean to wash clothes, linens etc. either by hand or with a washing machine. One does not exclude the other.
Laundry refers to clothes, bedding, towels etc. and before the invention of the electrical washing machine, it had to be hand-washed, in other words it was laundered by hand.
One of the earliest and simplest washing machines was the scrub board, also called the rub board and washboard. Invented in the 18th century, women were no longer constrained to scrub clothes in a river using stones, or visit communal or public washhouses, they could wash their family's clothings at home.
With the invention of the electrical washing machine, women's hands were physically freed from doing laundry, but it didn't mean the responsibility or drudgery of laundering became obsolete. Linens still had to be dried, and later ironed.
The OP's sentence is therefore grammatical and semantically sound, but there is a final element to consider.
‘The washing machine’ vs ‘washing machines’
When we speak about inventions that are countable, it is common to refer to them in the singular, and to add the definite article, ‘the’, thus some might claim: "Man's greatest invention was the wheel" while others would state: "Fire was mankind's most important discovery". The use of the definite article with inventions is common but not obligatory, therefore the OP can write either of the following:
- Washing machines freed housewives’ hands from doing [the] laundry.
- The washing machine freed housewives’ hands from doing the laundry.
Examples of “doing laundry” and “doing the laundry”.
From the website The Great Idea Finder
Women from all classes tried to find ways to get relief from doing laundry. Some hired washerwomen and others used commercial laundries. Eventually mechanical aids lightened the load.
and from what-when-how
On the other hand, it was probably because the electric washing machine made the task much easier that American women, still primarily responsible for the family laundry, were able to pursue careers outside the home.