Well, I searched google for AD8620 counterfeit, problem, fake, I could only find 4 references, and 2 of them were on Head-fi, you were the author, and you had no direct experience, you were relying only on reports from other people. The other references weren't well supported, one at least being based only on a suspicion that the price was too low.
You have to be careful about starting or propagating 'Internet Legends' especially when substituting opamps doesn't work out as well as hoped, an ill-advised opamp change can easily degrade performance.
Consider that the AD8397 will drive 310mA into a 32 ohm load, where the 8620 is good for 30mA only, and that into an unspecified load. This is a factor of 10+ in only a single parameter, albeit quite a significant one where driving low impedances is concerned. A little arithmetic suggests that (11.8V/0.03A = 393 ohms) the AD8620 will only happily drive just under 400 ohms before the output voltage starts to become restricted, whether it simply falls off or clips is open to speculation. What is certainly true is that with lower impedance loads the distortion will rise, perhaps giving rise to the differences detected in your 'blind testing'.
As for the necessity of taking risks, surely it would be a lot less risky to just leave the original opamp in place and not risk damaging the PCB?