The central supermassive black hole does consume the galaxy, but in fact the rate of consumption is so excruciatingly slow that the galaxy will have "evaporated" long before then. Random processes like collisions with other galaxies will eventually eject nearly all stellar remnants into intergalactic space. The central black hole will only end up consuming about 1% of the galaxy's stellar remnants (as you say, all stars will have died trillions of years before the galaxy evaporates).

The Milky Way is fated for just such a collision with Andromeda in a few billion years, and there's a chance our solar system will be ejected!