Time, not space, is your answer here.
Two reasons -
(1) civilizations might not survive long enough to do this.
(2) 13 billion years is a long time. So you have the reciprocal of that as the chances to be in the right year to see such a probe. And with results from the new telescope we now have hints that the 13 billion number is bogus, the universe is likely far older.
What are the incentives to build and deploy such a thing though? We as a civilization fail to fund things that have a ROI of more than a few years, how are you going to fund something that pays off after a million year?
Consider that in some countries on Earth, we can't even get consensus that obtaining energy directly from the Sun via solar panels is a good idea.
First of all it's going to be massively more expensive than any housing we've ever built on earth so only a very small elite could afford living there.
But then again, space is a very hostile environment: it's super dangerous (any incident will almost certainly snowball into a dramatic accident), very unhealthy (billionaires are currently funding longevity research, so I don't think they'd like to go in a place where they would age up significantly faster than on earth…), and life is just worse up there on all respect…
"Extremely improbable" would be a better assessment.
Even ignoring the project complexity, difficulty, and energy budget, which can't simply be handwaved away by "robotic automation", one reason is simply that such colonies don't solve any problem that we're likely to have, that can't be solved much more cheaply, safely, and effectively.
But even the idea that we'll eventually have the technology to build such structures is debatable. Will this be before or after we solve climate change, for example? Because that issue is likely to severely impact our technological capabilities over the timescales involved. And as of today, the most technologically advanced nation is doubling down on atmospheric carbon production.
With sublight velocities achievable today, I recall it would only take around a million years for a Von Newmann probe to cover the entire galaxy. Such a probe is quite conceivable, so why isn't there more evidence of such probes everywhere?
Another point I feel is that proliferation of life should be an self-reinforcing affair, for intelligent life even more so. A spacefaring nation may terraform or just seed planets, and these in time will replicate similar behaviors. At a certain point, a galaxy teeming with life should be very hard to reverse given all the activity. A life itself isn't necessarily evolved from biology, AI machine lifeforms should also well suited to proliferate, yet we don't see them anyways.