If these demigods live 6000 years, their "childhood" could be 300 years long. The effort to maintain and educate such a "child" would put a strain on the parents, and while some would have more kids and some would have less, the "scale" of the lifespan would balance out.
Becoming a Demigod is dangerous.
Perhaps whatever method is used to achieve that long lifespan is inherently dangerous. You grow until 40 years old as a normal human would, but then the demigods go through a biological change which has a high failure rate. Maybe it's an external drug, maybe it's some sort of symbiosis with another form of life on the planet, or some chemistry in the environment, or maybe it's just the end of this species "puberty."
Surviving that long is dangerous.
Another option is that the ruling pyramid of the Demigods use younger demigods as tools, like western gods use angels. In order to make it to your first 1000 years, you have to survive being sent to the front lines of every holy war the demigods fight; surviving multiple trips back and forth to your realms, and your subjects learning that you're long lived but not immortal and trying to kill you.
Surviving your family is dangerous.
Maybe they DON'T do anything to reduce their child-rearing rate, but in order for a child to make it through to adulthood they have to live in a rough-and-tumble dog-eat-dog world, where only the strongest (and therefore healthiest) actually make it into the ranks of adulthood.
...and maybe this never stops, and even at 5500, you have to keep one eye open.