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I have played against a fair amount of AI's (Of all strength levels I might add) and I have never seen the AI use their starbase's ability to hold planets. Has anyone else experienced this?
I have, and I find it a bit disappointing - they should know to do this, especially for high-value planets such as Terran or Desert.
But possibly it is just about impossible to code and that is why it has not been done. The AI will also never use Wail of the Sacrificed as Advent Rebels or Stripped to the Core as Vasari Loyalists. If they AI could be programmed to use these later, it would be great, but as far as I know the devs have no intention of it at the moment.
It is actually technically possible for the AI to get auxiliary government now. It used to be that the AI would prioritize combat upgrades (weapons, armor, strikecraft) above all else and never get any other type of upgrade. Some patch ago that is now changed, I have personally seen AI starbases have auxiliary government (once) and meteor storm a couple of times. However, it seems pretty random as to what non-combat upgrades the AI receives, thus its a pretty small chance it will pick auxiliary government because the AI needs enough cash to research that particular upgrade as well as to purchase it. And even if the AI gets all these things, its not going to be smart enough to realize it should put AG on its homeworld instead of an asteroid.
In short, the AI can get it, but it is rare and even rarer for it to use it effectively. I asked in the Rebellion beta to hard code the AI to get auxiliary government on its homeworld as soon as it can afford it, at least for capital victories, but we can see where that went. 
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