It makes a lot of sense that Sanae would focus on the kids of the kingdom since that is all she knew in her previous life.
Looking ahead, it might be tough to convince people of germ theory. It took ages for the idea to be commonly accepted in our society. It really was the microscope and direct observation of microorganisms that managed to convince people.
A prince/king may have the authority so people will believe it though.
But even if they don't, if the orphans get healthier just by washing more often, maybe the idea that being clean is good spreads out more, regardless of germ theory.