I don't think we adopted handwashing because of him.
Actually, the reason people didn't believe him is interesting because they were using an earlier scientific method - he didn't have a good explanation for the mechanism, so they dismissed it with "correlation is not causation". We're better at using things whose mechanism we don't understand now, like Tylenol.
There is a whole industry in off-label use of medicine, and there are also quite a few studies that try to move off-label use to on-label use once the link is clear.
What is interesting (to me) is that which we now consider to be common sense was at one point controversial.
Actually, the reason people didn't believe him is interesting because they were using an earlier scientific method - he didn't have a good explanation for the mechanism, so they dismissed it with "correlation is not causation". We're better at using things whose mechanism we don't understand now, like Tylenol.