The problem will not solve itself if it's ignored. All the current calculations of future positions of course include the effects of the changes of the orbits.
The reached tipping point they mention is that the current objects will collide often enough to continually produce even more debris. That means that the generation of the new objects happens certainly much faster than the burning in the atmosphere.
"today, the catalogue contains about 13,000 objects, or more than 3 times as many objects. This gives a collision rate that is more than 10 times what it was just over 30 years ago, or 0.13 per year….which is the same as one catastrophic collision between cataloged objects every 8 years….with the time between collisions rapidly becoming shorter as the catalog continues to grow. The larger fragments from either explosions or collisions will further accelerate the rate of collisions."
Only if we stopped putting more crap up there for those few decades (where a "few" is probably more on the order 10 to 100 ... yes, decades). Human exploration and exploitation of space resources is far too important to the short and medium term benefit of mankind — leaving entirely aside its implications for our long term survival — to wait it out.
Or put it another way: the GPS system is used to synchronize power phases, guide aircraft, cars, people, ships. It's built into emergency transponders, military drones and bombs.
Without that system, we're back to compasses, line of sight and dead reckoning. We lose the current insane advantage of realtime positioning anywhere on the planet.
That's 1 18 satellite system - and we absolutely can't run a modern civilization without it.
Actually, nowadays a system of ground stations and high altitudes atmospheric floating balloons can be made to serve the very same purpose. The ground stations would serve for triangulation of balloon's position and those at their turn could serve the rest of the service-consumer base. As a bonus, the balloons would be obviously more manageable compared to satellites, the cost of taking them into operation would be of course much lower (no more atmospheric pollution), and be disposed cleanly without causing the problem discussed in the article.
I think your estimate is overly alarmist. Without GPS, we'd use ground-based stations. It wouldn't be as accurate or as available, but it would still be good enough for almost everything GPS does. In fact, we already have ground stations to augment GPS. And even without ground-based stations, modern accelerometers allow inertial navigation to be quite accurate.
Also, GPS satellites are in medium Earth orbit, which is rather sparsely occupied. In a Kessler syndrome scenario, GPS would likely survive.