Why are almost all tires today radial ply tires?

Because radial ply tires are better than bias ply tires.

The belts are layers of woven reinforcing fiber inside the tire under the tread. The belts give the tire strength, keep the tread from squirming and help resist punctures. The belts may be fiberglass, steel, nylon, rayon or aramid fiber.

In bias ply tires, which were common before radials were invented, the innermost plies crisscross like an "X" as they run from bead to bead (side to side), while the outermost plies run lengthwise around the circumference of the tire. This makes a relatively firm tread, but it also increases rolling resistance. And anything that increases rolling resistance increases fuel consumption.

The innermost plies in radial tires, by comparison, run sideways between the beads. This makes the tire more flexible, which reduces rolling resistance and improves fuel economy. Changing from bias ply to radial ply tires can improve fuel economy 10 to 15%, so that's the main reason why most tires today are radials.

The increased flexibility of radial tires also helps improve traction and cornering. Because the tire is more flexible, it is better able to maintain tread contact when cornering. This also allows lower aspect ratio tires (shorter sidewalls) which provide better handling performance.