Not in Japan but live in India which follows essentially a similar system. In India at least, the short answer is the employer doesn't.

However, every financial institution that deals with you deducts and prepays some taxes on your behalf and also reports the income. So at the end of the year, it shows up in a report which shows tax deposited and you just have to fill in the sum of that income as "other income". This system is good enough that if you are a salaried person with some investments and some tax deductibles like housing loan, tax filing takes about an hour and also normally doesn't have any surprise element of "you owe the taxman X"

It becomes complex only for business owners which is a small minority of the population of any country.

Japanese securities companies have special accounts which automatically report tax-related information to the tax agency and withdraw taxes appropriately. So you'd be able to do standard stock and currency trading without being required to file a return. It's quite nice being able to work at a company and participate in the market, and still not have to worry about filing.

Doing rarer activities such as investing in startups brings you into return-filing territory though, but I guess that's unavoidable since it's private investing, and not a concern for most people. Reading many comments on this post, it's clear hackers generally aren't "most people" ;)