Earlier quoted context omitted.
But isn't it? I mean low earth orbit is at least as much space as we have on the surface, which is already pretty damn big. Yeah, speeds and orbits and stuff change the game a bit, but i can't imagine even a few thousands of satellites cramping a space that large.
The problem is we want a lot of things to be in geostationary orbit, which is only possible at a specific altitude. That effectively transforms the "space" from three dimensions to two because the third relative coordinate for all of those things has to be the same. Which is when relative motion becomes a problem. The Empire State Building and the Trump Building are at approximately the same altitude (i.e. sea level)…
But anyway space is very big and the probability of collisions is still low. Two geostationary satellites out of control would probably not collide, they will pass a few hundred of meters away in any direction. The big distance between them is mostly to avoid interference between the radio signals.