To expand on what @tdelillo said, if your courses are using start and end dates, they should automatically get pushed into the "Past Enrollments" category when the course is over.
This will allow the Dashboard view to automatically update the tiles with current courses.
With respect to the Admin view, I understand what you are saying, but I am not aware of any way to tell Canvas to default to the current semester. I have always had to filter. However, I am fully integrated to the SIS and do not manually create courses so it is not an issue for us. If I have to use the Admin view it is usually to lookup a specific course.
If you have a "student information system" I highly recommend integration to automatically create courses and rosters.
This discussion post is outdated and has been archived. Please use the Community question forums and official documentation for the most current and accurate information.