Not exactly. Time does run more slowly near a strong gravitational source, and if you are near the event horizon of a stellar–mass black hole this effect can be extreme. However, the larger the black hole is, the flatter the space around it. Furthermore, “near” is relative. A Birch world would be built around a black hole which is approximately a light–year in diameter. The structure would be “near” the black hole’s event horizon in relative terms but in absolute terms it would still be pretty far away, perhaps a quarter of a light year. Expect a time dilation of just 2:1, meaning that for every year on the Birch world two years pass for the rest of the universe.
It might seem like this costs you a lot, since it halves the amount of time you can live near your black hole. However, the lifetime of that black hole will be somewhere between 10¹⁰⁰ and 10¹⁰⁶ years, which is pretty insane even if you only get to use half of them. Furthermore, this is many orders of magnitude longer than the lifetime of a galaxy, so your civilization could potentially outlive everything else in the universe. Large stars burn out the quickest, but with black holes it is the other way around: small black holes evaporate the soonest. You might think that storing hydrogen in brown dwarf planets for use in fusion reactors would power a civilization for a long time, but fusion reactors are surprisingly inefficient. A civilization built around a rotating supermassive black hole can take advantage of the Penrose process to extract more usable energy from the same mass than the fusion reactors would.
It might seem like this costs you a lot, since it halves the amount of time you can live near your black hole. However, the lifetime of that black hole will be somewhere between 10¹⁰⁰ and 10¹⁰⁶ years, which is pretty insane even if you only get to use half of them. Furthermore, this is many orders of magnitude longer than the lifetime of a galaxy, so your civilization could potentially outlive everything else in the universe. Large stars burn out the quickest, but with black holes it is the other way around: small black holes evaporate the soonest. You might think that storing hydrogen in brown dwarf planets for use in fusion reactors would power a civilization for a long time, but fusion reactors are surprisingly inefficient. A civilization built around a rotating supermassive black hole can take advantage of the Penrose process to extract more usable energy from the same mass than the fusion reactors would.