Read on Twitter
OK, I'll bite. But I think there's a problematic premise behind the question. Let me start there. https://twitter.com/JapanIntercult/status/1262721293549309953
Many people think, reasonably enough, that the things we tend to call "religions" operate kind of like computer programs or operating systems that drive humans as if we were robots. I don't think that's the case.
Empirically speaking, we have no way of saying that any "religious" idea serves as the prime motivator for behavior. What we have are ex post facto rationales that people offer for their behavior. Some may say they did X for religious reasons, but we'll never know if that's true.
That's not to say that cultural (for lack of a better term) influences on individual behavior don't exist, but it's hard to pin such diffuse factors down to "religion" specifically. (The religion/not-religion distinction is itself cultural, but that requires a separate thread.)
OK, to the subject at hand: Until recently, it was very common for textbooks on Japan to describe Shintō as a religion concerned with purity and pollution. Sometimes this is explained, other times it just sort of appears without further comment.
I think that there are two primary places where we can see this concern with purity and pollution, but before I get there let me offer a caveat: Japanese people were most definitely NOT operating according to a germ theory of disease until about two centuries ago.
(This is not my area of expertise, but Ann Jannetta's *The Vaccinators* might have more on when germ theory appeared. Been a long time since I read it: https://www.sup.org/books/title/?id=7697)
The first purity/pollution distinction people are likely to think of is a scene in the classic myths (Kojiki/Nihon shoki). Izanagi flees the underworld and then bathes to remove the death defilement. Several major deities are born as the process of his lustrations.
So in this scene we at least have a death=bad/defiling and washing with water=good/purifying idea. This is the origin of /misogi/, a type of cold-water ablution: https://tinyurl.com/y9xl6fuf
That, in turn, forms the basis for the abbreviated ablutions shrine visitors do at the temizuya (some say temizusha), or the hand-washing font found at the entrance to many shrines: https://tinyurl.com/yaf33xt4
(If you observe people carefully at the temizuya, you'll see various approaches to hand washing: perfunctory, deliberate, skipping the process altogether)
I don't think we can trace any direct line from what people do when visiting shrines to their hand washing practices after using the toilet, FWIW.
The second association between purity/pollution and Shintō lies in a longstanding practice of propitiating the kami to ward off epidemics. Here's the first article that comes to mind about this, although there are others: https://nirc.nanzan-u.ac.jp/nfile/2934
Notably, the Gion Festival (a famous festival that takes place every July except this year for obvious reasons) has its origins in efforts to ward off epidemics: https://www.gionfestival.org/history-of-gion-festival/
The Gion Festival became a sort of template for festival practice throughout the archipelago. While regional variations exist, elements of the Gion processions appear in many summer festivals across the country. https://tinyurl.com/yb66toos
Anyway, none of this tells us why there is usually no soap or towels in public Japanese toilets, nor does it tell us what Japanese people may be thinking when they reach for the soap and then go "Nah, I'm good."
I talked about Shintō here because that's what @JapanIntercult asked me to do, but hold in mind that there was no conceptual distinction between Shintō and Buddhism until ca. 15th c., and no effective split until the mid-19th c. So Buddhism is important here.
For some ideas on bathing in Buddhism, ask @_tobenedict or see this article by Max Moerman: https://nirc.nanzan-u.ac.jp/nfile/4401 . Buddhist ideas about karmic defilement and salvation were intimately linked with bathing practices (bathing as in whole-body immersion)
There are lots of stories about bathing and miraculous cures that are rendered in a Buddhist idiom. But again, nothing that would tell us much about contemporary hand washing practices.
At most, these stories tell us about how some contemporary onsen (hot springs) became known for their healing properties. Onsen today often tout their ability to heal specific maladies, and they do so in a mix of biomedical and "religious" (i.e., empirically unverifiable) speech.
That's all I've got! @bryandaniellowe can offer more on the ritual purifications people would do in ancient Japan, but I think he would agree that there's no direct line from that to contemporary hand washing.
