For cleaning tires, you need an alkaline based chemical to be effective in stripping built up grime. Shine Supply Wise Guy and Brake Buster are both alkaline cleaners and do this job very well, as would strong APC dilution.
My go to tire cleaner is Brake Buster, un-diluted for full cleaning power.
Honestly, I like both Adam's and Brake Buster but the stiff brush (or similar agitating device,) is the key component for removing tire shine. Honestly though, I try to preserve my tire shine and I avoid using a brush on the tires for a couple washes after tire shine to prolong its life. And I only use the wheel cleaning solution if I'm going to reapply the shine. Otherwise, I just go with DFB's recommendation to use nothing stronger than regular soap in the cannon for the wheels and tires.
Long-time Simple Green user but I learned the hard way to dilute it to at least fifty-fifty. Otherwise, it can pit metal fairly quickly. One of the best handgun cleaners out there, too especially if you use an ultrasonic cleaner. Good for small car parts only though.
Don't use SG on any parts of the vehicle however, I prefer wheel/tire cleaner in a foamer. But for nearly every other cleaning chore, SG is about as good as anything, I think. And cheap. I also like Zep Citrus Degreaser which is a very good non-petroleum based grease remover, too.
I use "Super Clean" non diluted, with a MaxShine stiff bristle brush. Takes about three applications until the foam created by the brush becomes white.