Yikes, the way I praise this stuff I almost feel guilty when somebody here tries it and isn't happy!
IMO you oughta try a different medium. Rather than the sponge, I use something absorbent, like an old cotton towel. Yeah, it gets trashed (or at least permanently stained in a horrible way) but that's what works best for me. I've done OK with MFs, but better with cotton.
You need two things here- 1) somewhat aggressive scrubbing ability (more so than you'll get with a sponge). You can sometimes get by without this but probably not for something as nasty as tires, and 2) something that the "brown stuff"/etc. can soak into so you you get it out of/off the surface being cleaned (rather than just moving it around). IME sponges just don't soak it up well enough.
Getting a used tire genuinely "*black*" can be a challenge. Often they still don't turn out quite the way I want until I dress them. But the Rubber Prep does get them cleaner than anything else I've tried without drying them out. I recently got a Michelin tire for the Tahoe that had *terrible* issues with browning/etc., way worse than the other three on the vehicle. When I did the last big wheel/tire cleanup (coating the wheels, scrubbing the tires properly) I used Rubber Prep on it and I can no longer tell which tire it is by simply glancing at the vehicle; before it was "that ugly tire is the new one" after no time at all.
I don't use Rubber Prep for tires nearly as much as I use it on other stuff, but I do think it works better for me than it has for you.
Give the cotton a try and see if that works better. Almost forgot- sometimes a given tire will hold onto cotton lint and when that happens I do the final passes with an old MF. Yeah.."passes" plural, I do go over tires quite a few times before all the nasty stuff is off.