The yo-yo quilts that I inherited from my grandmother (1940's?) did not have backings. I believe they were made to lay on top of the top sheet on a bed. This would then allow the sheet to peek through the areas between the yo-yo patches. From what I have heard, yo-yo quilts were purely decorative and usually not used as a blanket.
Of course, you can do whatever you want with your yo-yo quilt - it is your quilt. I like the blue backing (I know it was a sheet on your bed, but it looks good!). I don't have any idea how you would attach the top to the backing. I have seen sheets used as backings (many QB folks use them) but I don't believe they are any "heavier" in weight than yardage. I may be wrong about this.
Good luck with what you decide! I love yo-you quilts and is on my "bucket list" to make.