Assumption #1 is quite correct - there is a very large ("supermassive") black hole in the center of our galaxy.
Assumption #2, however, is false. Black holes are no better at drawing in distant objects than any other thing in space with the same mass would be. If you collapsed the Sun into a black hole right now, the Earth's orbit would not change. That is because gravity is gravity - it doesn't matter that we are being pulled by a star or by a black hole; all that matters is the mass of the object doing the pulling.
So yes, anything that falls into the black hole in the center of the galaxy is lost. But there isn't terribly much falling in. True, the large number of stars at the center tend to be moving in all sorts of random directions, and some inadvertently get sent on trajectories into the black hole, but most objects in the galaxy are unaffected. In fact, our Solar System is held in its orbit far more by the mass of all the stars near the central black hole than by the black hole itself, as these collectively outweigh the black hole by quite a lot.
Part of this misconception I think is due to artists' renditions of accretion disks where all the material inescapably spirals into the black hole in the center. However, the reason this material spirals inward is that gas can transfer angular momentum via collisions. Stuff moving in a nearly circular orbit has too much tangential velocity to just "fall" in. What happens is that material further away in the disk is moving slower, and frictional drag slows down the stuff toward the inner part of the disk, allowing its orbit to decay. Basically, angular momentum (essentially the tendency to have a tangential rather than radial velocity) is transferred outward as matter is dragged inward.
With stars, this process basically cannot happen, simply because they are not colliding with one another. Our Sun will continue for many billions of years in a stable orbit around the center of the Milky Way, black hole or no. Stars do scatter gravitationally (a process called dynamic relaxation) so there is an analog to the viscosity of an accretion disc, but it's much smaller and slower.