Hi @vkl86,
I think there's an important distinction here. What Canvas does keep track of (somewhere) is when an enrollment was first sent to Canvas. In some cases, this day/time may closely (or exactly) coincide with then they "registered" for the course, but that is definitely not universally true. In most cases, registration happens in a Student Information System, and then the SIS passes the enrollment instruction to Canvas at some point. Maybe this is instant, maybe it's not actually done for a couple months if the course is way out in the future. If you're having students use a self-enroll Canvas link, that info would be pretty accurate.
I say all of the above as potential reasons that the info may not be good to display to teachers. Again, for certain scenarios it might be great, but in others (which would be a vast majority for us), the information would be inaccurate and very misleading.
Now I'll close with a potential solution... It does appear that if the teacher role is given the "Users - manage students in courses" permission, they can actually see the enrollment details, though its multiple clicks to get there. Unfortunately, that permission unlocks other things (like the ability to conclude an enrollment) that may not be allowed at your institution. Might be worth checking on though.
-Chris
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