That is a really beautiful yo-yo quilt!
I would probably hand tack the yo-yo top to a sheet. The sheet would need to be a low thread count (or an older, used, lightweight sheet) to make hand sewing through it easier. High thread count sheets can be hard to hand sew. It would be fairly easy to hide a thread knot in a yo-yo, tack down the yo-yo, then carry the thread through a yo-yo to the next tack. You wouldn't necessarily have to tack every yo-yo down, as tacks every 6" or so would be enough to relieve stress on the yo-yo joins; however, it would probably be pretty easy to tack every yo-yo join since they seem to be joined in rows. I would not go through the centers. It would be easier to just tack at the joins.
While machine tacking seems like it would be faster, I think it would not look as good and -- because you would have to cut threads on both top and backing side between each tack -- would actually be more work and possibly provide less stability (since once you cut the threads, some tacks might eventually unravel). It seems to me hand sewing would be fairly easy to do in front of the tv.
The trickiest part for me would be basting the yo-yo top to the sheet, as you don't want anything to shift while you are tacking it down. Basting spray would not be a good idea if hand sewing because it creates way too much drag on the needle. You might actually need to hand baste the top to the sheet before doing the hand tacking.
Binding could be a problem. I think I would just machine hem the sheet all around about an inch away from the yo-yo edges, and then turn the sheet under to create a straight edge and hand baste the yo-yo edge to it.