Exactly. Tires do have alot to do with it, but body roll and spring stiffness also plays a factor in how much under/oversteer you get. Bigger front bars make the front stiffer, but more prone to push once tires lose traction. Whichever side you add a bigger bar to, you're going to get less body roll. That means less energy is wasted in body roll, and more energy is transmitted to the tires, and tires can only do so much (even if you are running 315s up front). This is why I and many others feel that 35mm solid is about as big as you need to go up front, and most prefer a 35mm hollow bar.
Tires are almost always going to be your limiting factor, but which tires lose grip (front for understeer, rear for oversteer) and how the car behaves once that grip is lost is a product of the rest of the suspension setup. More meat helps, but it isnt the answer for a poorly setup suspension. A 35mm hollow front bar, a good set of revalved Bilsteins from Strano coupled with your springs of choice (whether stock or aftermarket) will get you started and is a tried and true foundation. After that, just adjust rear bar size up if you feel the car is understeering too much.
FWIW: My car has had quite a few setups under it, and right now I dont even have my front bar connected for drag racing purposes, but I normally run a 32mm/19mm bar combo with Koni SAs dialed up about 60% in the front and 10% in the rear, all on stock springs with 275 Kumho 712s all around. It's a good all around setup thats main goal is good street manners that can also work decent at the drag strip and autocross course, with very little understeer/oversteer bias (enough of a balance that on the streets I cant tell which way it falls, and I haven't autocrossed with this setup yet to be sure, but I think the car has just a touch of understeer, which it should, as the gas pedal can balance that out in short order), but I trade off and take more body roll for a bit more comfy ride and that kind of balance.
