I'm not fond of the ingredients in those cleaners. Mother's Rubber and Tire Cleaner is a little less harsh as is Michelin Tire and Wheel Cleaner. Yep, need a good, stiff dedicated white wall brush.
Just purchased some wide whitewall tires for a classic car. The guy told me to use just soap and water and to make them shine dig this, use mop & glow I kid you not that was what I was told.
Ah yes........I'm old enough to remember doing my tires with mop and glow all the time in the days before there were dedicated tire products. It actually worked pretty well but most of the dedicated products available now work much better.