I could not find a reliable source that ascribes Japan's relative success to masks. Some ideas are discussed in https://www.bbc.com/news/world-asia-53188847 . The advice from Japan's ministry of health on masks is not a ringing endorsement, either: https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/kenkou_iryo... .
We aren't certain that it's masks. But they did little else (e.g. no lockdowns or travel restrictions); so it's either masks, or something unknown. Occam's Razor says masks are more likely.
There are other anecdotal cases about mask efficiency. E.g. there is a hair salon in US where two hairdressers got COVID, and they served 139 clients in 8 days, before testing positive; one was symptomatic for the entire period, the other one for 5 days. Per company policy, they wore mask consistently while working with clients.
During that time period, they infected 4 people - but all of these were their family or roommates. None of the clients they serviced developed symptoms then or within a month, and of the half who consented to testing, none tested positive.
Besides, we do have good reasons to believe that masks are efficient just based on physics and biology alone. We do know that masks are definitely effective at stopping saliva droplets, and we do know that droplets definitely spread COVID. So even if it has other vectors (e.g. aerosols), masks would still reduce the overall viral load, and thus chances of infection.
"Stay home" is one of those things that are on every list of recommendations, but it's completely non-viable in practice because of the way US handles healthcare in work context. Keep in mind that there is no federal requirement for paid sick leave, and only 13 states mandate it. Consequently, 30% of workers don't get any paid sick leave at all - mostly the lowest-paying jobs, that are paid the least (and so taking an unpaid day off can easily mean an unpaid bill etc), and who interact with random people the most.
And don't forget that employment in US is "at-will", meaning that businesses can fire workers without giving a reason. FMLA limits that somewhat wrt unpaid sick leave, but 1) it only applies to businesses with at least 50 employees, and 2) it only applies to illnesses that are incapacitating to the person claiming sick leave.
The focus is on mask wearing right now, because the return to stay-at-home is clearly not happening, and so people go and mingle. If they mingle with masks, they definitely reduce the transmission rate enough to notice - there's ample evidence of that. What's unclear is just how large the effect is.
Also, why focus on Europe alone? We could also look at Japan, which seems to have made a bet largely on mask wearing - and has some impressive results to show for that.