My understanding is the only way to currently do this is by changing the term dates. I would prefer the term dates in Core be correct, but be able to adjust the access dates.

The problem lies when a teacher has a second semester class but did not have a first semester class so they cannot get to the drop down. They have to use a roundabout that is a bit cumbersome!

I think this is possible, we rolled over classes in the spring so that the pages existed, then gave access to teachers via Group Page Access in Core. Teachers just have to "view other classes" and filter by academic year to see their classes for non-current years.

This would be PERFECT! 

It's a little awkward setting the terms to non-term dates, but it works. In the end, is there any real advantage to just defaulting the view to All Year?

We do the same for the fall; our fall term starts on 7/16, same as our year.  But it's still a problem for back-to-back terms because the previous one must end first.  Just as the terms have a "Days to Default" where they extend beyond the end, I'd love an option to have it begin showing in the list of classes a certain number of days before the term starts.

You can do this.  I opened my teachers' classes on 7/1/15.