But the ringing, sustained oscillations, and excessive gain are sometimes desired characteristics of a particular op-amp implementation.
You will hear the effects of this in many hard clipping distortion circuits, though, where the amplifier gain factor will far exceed the voltage rails and be pushed into undefined clipping territory. Behaviors in this range can be an important part of the sound, e.g. the Proco Rat and the infamous LM308 op amp with its slow slew rate. Some like the TL072 exhibit a really nasty phase inversion that results in a pretty horrific (usually undesired) distortion.
It's a balancing act, though; search "op amp motorboating" in any DIY stompbox forum and you'll find thread after thread of people trying to keep op amp gain stages from oscillating. I know more than a few noisier artists who enjoy when designs can be tortured into doing that, though :)
There's a lot of constraints that either remain due to the restrictive power supply requirements on top of tradition or people building off designs from the 90s that were hand assembled and took BOM golf to the next level.
The op-amp oscillation the article is talking about occurs in the radio frequency range. Tiny parasitic capacitances induce 180° phase shift at some very high frequency where the op-amp's open loop gain has dropped to unity.
You can't hear this directly since it's not in the audio range. It brings about distortion. The amp may pass audio signal, but the oscillation prevents it from performing properly.
I wasn't trying to assert anything about the article, just responding to a comment about desired op amp imperfections in guitar effects. It's fairly common practice to bandwidth limit high gain op amp stages to avoid the issues discussed in the article, amongst other things.