Will airplanes ever collide in the air?

Why do airplanes almost never collide in the sky? If you look at radar screens, you can see thousands of flights happening around the world at the same time, yet collisions are extremely rare. First, every aircraft is constantly monitored by air traffic controllers. They manage altitude, speed, and routes and separate planes in advance to ensure safe distance between them. That's why even if two aircraft look close on radar, they are actually flying at different flight levels and never intersect. Second, every modern airplane is equipped with a collision avoidance system. It continuously scans the surrounding airspace and warns pilots if another aircraft gets too close. In that situation, the system immediately gives instructions. One plane climbs, the other descends. So they move into different altitude layer.