It won't turn the Wii on, and it won't sync with the red button. However, it works just fine with Wii-Sports when I have the option to press 1&2 to activate it.
That really doesn't make any sense. The fact that it won't boot the Wii means that the controller isn't synced with the device properly, and pushing both of the red buttons at around the same time should've solved it.
It sounds like the controller is bad somehow. Only suggestion (short of getting it replaced) would be to replace the batteries, but you already said you did that.