Countries can still opt to use an exonym if it pleases them, and when the local name is not in written in the Latin script a romanization is used, so Japan in y

Countries can still opt to use an exonym if it pleases them, and when the local name is not in written in the Latin script a romanization is used, so Japan in your example would not be 日本, but Nihon or Nippon. As it stands, the Japanese prefer Japan, so it will remain Japan for the foreseeable future. Türkiye chooses otherwise, and that is fine too (although whether the change is just for diplomatic use or broader depends on how it is promoted in the decades to come).