At this point, Musk seems to be denying use of StarLink to hit targets inside Russia, apparently fearing that that would cross Putin too much and threaten a direct war with NATO, or at least Russia targeting StarLink satellites for destruction.
It is not clear to me that the U.S. government would do anything differently if it had direct control. It might even do less for Ukraine if it was fully under government control. Just like Russia has been using the Wagner Group for government purposes while claiming to not have direct responsibility, the U.S. government can help Ukraine's military resist Russia by paying for StarLink services inside Ukraine, while not directly targeting Russian assets with U.S. military assets.
But, there is always reason to fear that something you need is under the control of somebody else who might put other interests ahead of your interests. So, I can see why Russia, China, Europe, and some others would want to have similar satellite capabilities directly under their own control.
The real issues are going to be how many satellite swarms can we handle orbiting the same Earth, and whether some military will decide to destroy one of those swarms for military purposes in some future conflict, which, by presently available methods, would make for a very dangerous orbital environment for astronauts and even automated satellites due to the debris that would result.