The CDC has, for more than a year now, been saying that washing hands is a distraction from aerosols which are the "real" cause. That might have been true in 2021 and might still be true now. But whether the CDC recommends for or against it doesn't actually indicate very much about whether it reduces the spread of COVID, because the CDC has been erratic since the beginning of the outbreak. So that doesn't count as evidence either way.
Not sure what the answer to the specific question is, but there is solid evidence that cleaning your hands prevents the spread of HFMD, Adoneviruses, E. Coli, Salmonella, and many other germs. Independent of the evidence for Covid in particular, regularly cleaning your hands is a great idea.
This is also why P100 masks are recommended; they're really well known to filter out aerosols in general, much more effectively than N95 masks (which were originally intended to keep R-values below 1 or closer to 1).
I live in Japan now.
Japan has been very successful in dealing with COVID, even if the cases are now on the rise, the number of dead people relative to the size and mean age of the population remains very low compared with other countries.
I guess there are different factors that can explain this, one being that people used to wear face masks in Japan well before COVID, which made it possible to keep the number of cases low until vaccines were available. However, I am starting to find annoying the social ritual of cleaning your hands. My impression is that there is little to no evidence that washing hands does anything to prevent the spread of COVID and that people keep doing it because it has become a cultural norm. Am I missing something important? I don't want to criticize something that might have some value and I am just not aware of it.