Well, let me tell you a story. Back in January 2006 I traveled to Japan as a student. Due to a snafu I didn't get my Certificate of Eligibility in time to apply for a student visa in time for the program to start. The school told me to fly to Narita anyway and show them the COE. I was taken to a back room at immigration and held for an hour or two. Eventually an immigration officer asked me to write a written apology for not having a visa, which initially pissed me off because I was simply following my school's instructions. But I went along with it, and promptly got a one-year admission stamp in my passport, called Special Landing Permission (上陸特別許可). So yes, they have discretion.

And all the comments about Japanese xenophobia are unfounded. The reason for all these entry restrictions was that Japan was trying to cap the number of people coming into the country from overseas as a way of stemming the inflow of COVID cases until a critical mass of the population was vaccinated. Japan couldn't restrict entry by nationals because they have a constitutional right to come home. So they shut down entry by foreigners and then re-opened it in stages. To the extent that Japanese people complain about foreign tourists at all, 95% of their complaints specifically pertain to tour groups from mainland China.