In reference to the BA story: Do black holes really exist?

I took a look at the Wikipedia entry on gravitational time dilation, and I think I can see why the event horizon can never form, or at least how it's impossible for an object falling in to ever reach it. That vertical asymptote at the Schwarzschild radius is easy enough to see in the equations.

I still have two questions, though:
1. The article refers to this as a paradox - what makes it paradoxical?
2. How is it that people thought quantum mechanics might circumvent this apparent problem?