Teachers should be able to create an assessment, publish it so that students can see it on the assignment center, and have it automatically be accessible to take when the assigned time starts.

Currently, if it is both published and ready for taking, the students can access it early. The assigned time is pointless if that's the case. The assessment shouldn't open until the "assigned" time, even if it's published

The only way to make an assessment automatically ready for taking at the start of the assignment window is to make sure that it is not published until the assignment time/date. However, in this case the students cannot see it on the assignment center ahead of time. Otherwise, if it is visible/published, the teacher has to manually mark it "ready for taking" at the start of the assignment window / period.