The Supermassive black hole in the center of the galaxy only contains a tiny fraction of the mass of our Galaxy, so it has minimal direct effect on the orbits of most stars. However I have sort of one potential issue. Newton's shell equations shows that when inside of a spherical planet, you can ignore the effect of all mass further from the center then you are. And while the galaxy is shaped like a disk instead of a sphere, that doesn't help, because the net force on a object "inside" of a ring or hollow disk is outward. Gravitational field of thin 2D ring - numerical simulation
As a result this means that if you were to take a galaxy and remove all matter within x light years from the center, star sufficiently far out would be initially be affected only very slightly, but stars only slightly further would experience minimal gravitational attraction the the center, and possibly even experience a net repulsion.
I've concluded from this you simply can't have a galaxy where EVERY star has a very low eccentricty orbit without having somekind of large black hole in the center.
So considering this, what would the net result of the removal of Sagittarius A* be on the core regions of the galaxy in the long term.