The Kessler Syndrome: Are We Trapping Ourselves on Earth?

TL;DR Summary
Space junk is accumulating at an alarming rate. What happens if a chain reaction of collisions destroys our orbital infrastructure?
The Problem in Orbit
Since the dawn of the Space Age, humanity has launched thousands of rockets, satellites, and probes. Many of these objects, or their discarded stages, remain in orbit. Today, there are over 30,000 pieces of trackable space debris larger than a softball, and millions of smaller pieces.
The Chain Reaction
In 1978, NASA scientist Donald Kessler proposed a terrifying scenario. As the density of space junk in Low Earth Orbit (LEO) increases, the likelihood of collisions rises. A single collision between two large satellites can produce thousands of new fragments. These fragments can then strike other satellites, creating a runaway chain reaction of destruction.
Trapped on Earth
If the Kessler Syndrome reaches a tipping point, Low Earth Orbit could become impassable. We would lose GPS, weather satellites, satellite internet, and the ability to safely launch humans into space. Humanity could effectively trap itself on Earth for generations until the debris naturally de-orbits.