The government owning AI companies is problematic because it opens up the possibility for market distortions that are long term bad for deflationary pressure on the cost of intelligence. Could be catastrophically bad if your goal is for the economy to speed up. Might be fine, but it's hard to predict. It's especially risky because if the logic is AI becomes economy then government owning AI means the government is also the economy, which has historically been quite an unstable amount of power for governments to wield. See the 20th century.
If the government wants to "give the entire American public a stake in the companies", which is a fine goal on its face, then there should be a deal where the shares are distributed and locked to individuals. Not intermediated by cronyism in some constellation of unaccountable investment firms, and probably not in what would likely be a highly poetically contentious sovereign wealth fund.
If the government wants to help the AI companies win, which is a great goal on its face, then they should give them competitive lines of credit, particular in the event that alternative sources of credit start to dry up, or if there are cyclical pressures on credit that make the next scale up steps too hard or slow. This is still market-distorting, but it's less risky and we shouldn't lose sight of the goal -- America and Americans winning and being better off thanks to the AI revolution.
