Foreigners were only allowed to be at a few designated ports, right?, and the Jesuits just... walked away?, no one stopped them?... and then they got a house at a town and started just... living there?... no one alerted the authorities?, no government official came to check on the group of foreigners living illegally in the country?, and then they even published a book about the Japanese language?!... I mean, cool for them but... weren't they at least trying to keep a low profile or something?
I'm so confused and wikipedia does not help
I believe you got the timeline wrong. Jesuits were in Japan from 1549. The Edo Bakufu only banned Christianity starting in 1612. Foreign ships were only restricted to Hirado and Nagasaki from 1616 onwards[1][2], and all missionaries were banned. Jesuits did keep trying to sneak into Japan for some time afterwards, but most of their missionary activities, and books about Japan, were from the latter half of the 16th century. In the 17th century, the Bakufu very much did round up and execute any missionaries they found in Japan.