I've said it before and I'll say it again: despite the impressive advancements in current LLMs, there is absolutely zero evidence supporting the notion that these AIs want to do anything at all. They simply respond to prompts. They don't even remember anything, the entire previous conversation needs to be included in every prompt. There is no technical mechanism by which they are able to possess or attain agency.
My best bet for the explanation of the Fermi paradox is simply the intersection of economics and interstellar travel timescales. The payback time for any "interstellar colonisation" is measured in centuries or millenia, and will require tens of thousands of people to condemn both themselves and their children and grandchildren to never see anything else than the inside of a spaceship. The only factor extreme enough to motivate humans to such endeavours is impending planetary destruction, which does not result in anything like exponential growth.