Pogo, I have no doubt that magazine writers say things like that - but it doesn't mean that it is true. I remember reading a magazine article decades ago, which said that a ring placed around the base of a pole or cable extending from Earth's surface to beyond synchronous orbit altitude would experience "cosmic lift" and rise to the top without being pulled by any cable. Anybody who thinks that is true would never pass my old junior high school basic science class.
So, thinking about "Kevlar can support its own weight" in realistic terms, how much Kevlar, how much does it weigh, and how does that compare to the measured tensile strength of Kevlar? I am willing to bet that you can't make a 22,000 mile plus Kevlar cable that can support its weight when hanging by one end. But, it is a complex calculation, due to the change in "weight" of the segments as they are at higher altitudes with higher "centrifugal force" counteracting gravity. So I am going to leave it to the people who want to convince me to do the work to show that it does support its own weight.
While I am at the debunking business for this concept, I should also point out that the idea of having an extremely tall tower based on Earth's surface needs to consider the response of the Earth's surface to the extreme weight of the tower. The force on the base of a tower 22,000 miles tall would likely be enough to drastically deform the surface. If the base is broad enough for the base to not break through the crust, sink into the mantel and start melting at its lower end, then it would bow the crust downward in a very substantial manner, which would probably induce substantial earthquakes, which would produce lateral loads on the tower structure that probably would cause it to collapse.
Think about it this way, ice sheets on land are known to cause Earth's surface to sink towards the center by substantial amounts, and those are sheets of material with densities less than 1 gm/cc and not more than 2 miles thick. What do you think would be the effect of something that is more than 11,000 times thicker and more dense? Even Kevlar is 1.44 gm/cc, and steel is in the range of 7-to-8 gm/cc. The forces would be enormous, whether for a small base of extremely incompressible material (diamond maybe), or a wider, and thus more massive base.