No. The fact that galaxies rotate is thus explained, but the sped at which stars revoke about the central black hole can’t be explained by Newtonian gravity or GR. Basically, the speed depends on the mass of the central body. The reverse is true; knowing the orbital speed allows calculation of the central mass, assuming the orbital distance is known. The mass calculated this way should be independent of which orbiting body is used. For galaxies this is not the case. Stars further out orbit faster than predicted. This is only possible if there is unaccounted for mass or GR is wrong. Hence either dark matter or MOND. Dark matter works better in cases other than galactic rotation, so that’s the accepted explanation.
I get what you are saying, but I think you are describing two body dynamics, which work for the Earth orbiting the sun or a satellite orbiting a planet. I wonder if in the case of stars on the edge of a galaxy orbiting the galaxy, you have to take the sum of all the stars inside that orbit and include it in GM. In other words its a multi-billion body problem and the dynamics are much more complicated.