Is the ability to stop the car affected by anything besides the tire grade?
Yes. When the tread is badly worn, all tires lose much of their stopping ability, especially on wet roads.
For safety reasons, it is important to replace tires when the built-in treadwear indicators begin to show (you will see lines across the tread when the groves measure only 1/16 of an inch deep (about the thickness of a nickel).
Regardless of the traction grade a bald tire is dangerous. Keep in mind that properly maintained brakes are also extremely important to stopping performance.