"Additional attempts" relates to opening up separate attempts, where the previous attempts are kept as a history of submissions.
As students aren't required to mark their assignments as submitted, their existing attempts are still open. So, when you grade their attempt, they are not granted a new attempt, but neither are they prevented from continuing to edit their current attempt.
If you don't want students to have to mark their work as submitted, then you'll have to use the lock option (on the overall grading page) to prevent them changing the submissions.
It would probably be good for Moodle to add an option to lock submissions when graded, but the code doesn't currently support that.