Central black holes' spins don't drive their host galaxies' rotation.

* Elliptical galaxies exist where the rotation curve almost exclusively measures radial motion of blobs of gas (radiating specific wavelengths, whose redshift we can compare with other parts of the same galaxy), and sometimes other bright objects with characteristic spectra, which move outwards and inwards on an elliptical orbit; these galaxies may not spin around an axis, and may not even have anything like an equator. The anomaly is that the difference in these rising-and-falling orbits' is not as large as expected -- the outer reaches are dimmer, less dusty, and less gassy than the denser inner reaches, so there is less ordinary mass out there, but this reduction in mass density is not reflected in the orbital speeds;

* In discoid galaxies (spirals, etc.), the spin-axis of the central black hole may point at any angle relative to an axis through the galaxy centre, perpendicular to the thin disc; the polar jets of black holes can even blast mostly into the thin disc, rather than mostly out of it; the polar jets are indicative of the spin-axis of the central black hole;

* Galaxies may have more than one central black hole, especially the larger ellipticals. These central black holes typically do not have their spin axes point in the same direction, and where the spin axes are remotely close to perpendicular the black holes might not all spin in the same direction;

* Some central black holes are gargantuan, some are petite (ours is small, especially compared to the number of stars in the Milky Way), and some are apparently absent, but the rotation-curve anomalies appear to be very similar nevertheless.

That said, active central black holes -- "Active Galactic Nuclei" (AGN), of which quasars are a species -- can blow tremendous amounts of matter out of the central regions of their host galaxies, and that can alter the orbital speeds of gas and dust clouds in those galaxies generally, and much more so the behaviours of X-ray-bright gas and dust they shoot outwards into extragalactic space. AGNs can switch from very bright to essentially off, and as far as we can tell the switch-off of AGNs does not cause the rotation curve anomalies to relax. (Our extremely quiet central black hole may have been an AGN perhaps as recently as millions of years ago. <https://en.wikipedia.org/wiki/Galactic_Center#Gamma-_and_X-r...>).

Frame dragging by the rotation of central black holes in AGNs may be seen in the corkscrewing of their jet emissions: Galaxy 3C 348 shows this clearly <https://duckduckgo.com/?q=3C+348+galaxy&iax=images&ia=images...> as does the more popular M87 <https://apod.nasa.gov/apod/ap011101.html>

These jets are very bright in some characteristic wavelengths (radio in 3C 348, X-Rays in M87), but they're much sparser than the starry regions of their respective galaxies. These two galaxies are large ellipticals, and their jets do not align with anything like a central rotational axis as in a spiral galaxy.

The screw displacement related to frame dragging happens in the near region to the source black holes. Anomalous gas-cloud rotation curves are found in the far region, at the outer edges of galaxies.