A sufficiently advanced civilisation would be able to control their birth rate to whatever rate they would need at the moment. Think of an artificial womb (if that civilisation is mostly biological, if it's mostly robotic, controlling the birth rate becomes even easier).
It's also likely that a self sufficient, long-term thinking planetary civilisation would limit their population to a certain size, to not needlessly exploit the resources that they have available.
Of course, my point is that a high birth rate is something we can’t assume past a certain threshold of development. I can see the argument for being in two, three star systems - it gives you redundancy. Beyond that you add complexity with no inherent benefit.
In fact, the only motivation that makes sense w/o exponential population growth is a militaristic/defense oriented one, which would also favor the dark forest theory.
It's also likely that a self sufficient, long-term thinking planetary civilisation would limit their population to a certain size, to not needlessly exploit the resources that they have available.