Exponential growth doesn't inherently require population expansion. This would especially be true for a far more advanced civilization. You could dramatically amplify your colonization efforts with machines, reducing your need for traditional population expansion (productivity expansion in tandem with population contraction is exactly what we're going to see on Earth this century).

Without population growth you lose the motivation. Communication gets harder every time you add a star to your civilization, unless you somehow break the speed of light. Resource use is unlikely to be a motivator - a civilization sufficiently advanced to be multistellar will no doubt have cracked either fusion or the stain sphere problem.

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.

There is no evidence this will hold true in a colonization scenario.