Having suffered two failed attempts at securing a garage door opener that will actually fit, I was very discouraged.  But then I noticed that the neighbor had a garage door opener.  And since that house was the exact same floor plan, we checked out their garage door opener.

Yes indeed, they had a working garage door opener that used a different approach.  Instead of an overhead track with a motor at the end, it mounted the motor right at the door opening.

This certainly looked like the solution.  So, I started calling garage door companies.  And, yes, they were familiar with the product.  Unfortunately, it is no longer available as the manufacturer went out of business.

Well, back to square one.