I just experimented and got the same message. So you are right in part.
However, it is not correct that only NVIDIA supports Flash. The relationship is the other way around. Adobe chose VDPAU.
Adobe supports VDPAU. NVIDIA does not support Flash.
ATI cards will accelerate Flash in gnash and have since 2010, but I never had good luck with gnash -- but I haven't used it in a long time. Certainly not since 2010.
Edit: I installed gnash on a machine with an ATI card. It did, in fact, allow acceleration and put a really good load off on the GPU -- putting as much as a 30% load on the GPU. Unfortunately, I could not get 1080p content to play (the little spinning balls went on forever) and not all sites recognized it (on 3dgameman, for instance, I got gray boxes and no video.) So, it's not an inherent problem with ATI, but simply that Adobe has chosen to use VDPAU.