Ran my 2012 GT into the 10's with Stiff sidewall Hoosier 28x10.5 slicks in the rear and Hoosier 17" Radial skinnies up front. Hundreds of passes on that combo, in that car. Ran the rear tires as low as 10.5psi and that was the only time it got a little hairy, and it was ALL in the rear.
The next tire combo I ran was Goodyear 28x10.5's and M&H Bias Ply fronts. Again, no issues at the track but I preferred the Radials for Safety reasons. I have seen Bias ply fronts blow at the track. I'm a big fan of the 8 ply MT Bias fronts and the Hoosier Radials for these heavy cars.
IMO people are jerking the wheel when shifting and/or running too narrow of a wheel for the rear slick. That's where most of the sway problems we see come from, or simply too low on tire pressure.
I believe the OP was intending and referring to running a radial street tire. Not a radial skinny tire. This might be where some confusion is coming from. A skinny radial will not prove to be as unstable as a full size street tire.
I apologize for being intuitive enough to interpret his post and be able to give advice based on that, not just random facts or opinions that don't apply to his situation. He drives an M6 and was wanting to save money by not buying a front wheel/tire combo.