I see now. Looks like you used one of the first two methods "From the desktop" or "From the raspbian repository". I looked at the installed products from the Raspbian repo and this looks like the standard Debian armhf quake3 which links with Mesa libGL.
My guess is that the Raspbian repo used to provide the /opt/vc-based brcmGLES-linked RaspbianQuake3. That was probably the case back when the article was written, then at some point it was accidentally switched back to the standard Quake 3. This has happened before for packages like Kodi, or even VLC as recently as last week
Until someone ports Quake 3 to work smoothly with the open-source driver, you need to use the third method on that page "From GitHub" to build the brcmGLES-linked ioquake3. Alternatively, try q3lite
I know everything I said above sounds a little complicated, so feel free to ask questions so we can clarify.