We can never know when an asymptote will occur or if we're in one. There's simply not enough information as noise often makes it look like we're on an asymptote when in reality we are not. Asymptotes only exist in hindsight. By probability almost all time on a curve is spent NOT approaching an asymptote. So the bet with the highest probability of being true is that we are not approaching one.

>but never gets serious traction at solving novel problems or working in the real world(outside its core skill set; generating text)

This isn't true. Transformers now power self driving. I already stopped using Uber in SF. All my taxi rides are driven by AI.