Yes and no. You have to make a choice of whether to launch a crippling attack against either the Enclave OR the Brotherhood of Steel. Once you make that choice, you have finished the final plot mission of Broken Steel, but you are free to keep playing Fallout 3 to your heart's content, regardless of which choice you made. (Note, however, that if you turn against the Brotherhood, they will all be hostile toward you from then on.)
In fact, it has been observed that Enclave troops will continue to respawn after you're completely finished with Broken Steel, meaning that you can't actually "hunt down and eliminate the remains of the Enclave," which is supposedly the whole point of the add-on! Still, they make a good source of Energy weapon parts, so I'm glad to have them around.