First of all, the red sync button is visible thru a small hole in the battery cover and the jacket... you can use a pin-like protrusion on the new wrist strap to reach it. I don't have a picture of it, but you can see it in the manual for the new Wii U system.
Second, I always use my old systems to save wear and tear on the new system. I still have the Gamecube setup and play those games there even with a backwards compatible Wii. I've always done that.