It holds for a central potential. If most of the mass is concentrated at the center of the galaxy--which is where the brightness is concentrated--then it will hold.
The speed of a star around the galaxy is 2 pi times the radius (assume circular orbits), divided by the period. The period, per Kepler, is proportional to the radius to the 3/2 power. So the speed should be proportional to the -1/2 power. Instead, the speed is roughly independent of radius.
Wouldn't you need a solid rotating disk of a galaxy to perform that way?
In a solid rotating disk, the speed is proportional to the distance from the axis.
True enough, but brightness isn't necessarily a good indication of where the mass is. Spiral galaxies, in particular, have enormous amounts of dust in and between their visible arms. The dust in the arms both adds to the mass there and makes them dimmer. The centers are mostly dust-free. This was known even before accurate measurements of velocity distributions, and is accounted for when astronomers complain about "missing mass."
As a lawnmower mechanic, I can attest to the truth of this statement, just examine a Snapper rear-engine riding mower "transmission."