wait, what? Correct me if I'm wrong, but increasing width increases contact patch, which should increase grip regardless of direction. While height essentially acts as a gear reduction. Drag cars run wide tires and they don't turn?
Yes, but it's not so simple. Often times when we go wider we get taller by default. But a wider tire of the same diameter (and compound) may have less linear traction than it's taller brother.
For linear traction you want a greater front to back contact patch, or in other words, longer.
For lateral (side-to-side) you want greater width in your contact patch.
A taller tire with low air pressure can put a lot of rubber on the ground. To get a better picture of this imagine a rectangular shape of velcro. Let's make it 5 (w)x 7 (l). Actually, this can also be done with a piece of tape if you have that laying around... Continuing on... Drag this velco (or tape) across it's respective surface long ways. Now change it up so that it's 7x5 and drag it in the same exact direction as the you did before... What you'll find is that there's considerably more friction with the piece as a 5x7 vs 7x5. Now, this is the *nearly* the same idea on a car.... Imagine that a 205/45/17 is the 5x7 velcro or tape, and a 225/40/15 is the 7x5. All other things considered equal, the 205/45/17 will have more linear traction, while the 225/40/15 will have more lateral traction.
This is all because of the shape and direction in which the contact patch is traveling. So just going wider won't simply give you better traction at the strip... Two things play into slicks being so great; the compound and lower tire pressure. The lower tire pressure allows you to put a significantly longer contact patch down than street tires even though the tire isn't that much taller. This is the objective of lowering your tire pressure at the drag strips and why you see "dat wrinkle" in drag slicks. A wider tire with the same length (front to back) contact patch won't provide much more linear traction.
Also, I'm terrible at trying to get people to imagine things in their head.
TL;DR - Compound, tire pressure, and contact patch shape and size are help determine grip levels of a tire.
P.S. Sorry if this is erratic.