Of their debt portion, yes.
> is now trending toward the economy operated by quasi-government-owned companies?
No, as debt is not usually ownership (there are structures that allow it to become that). Companies just increasingly owe the government money, but that would happen under any traditional bailout scheme anyway.
> Is it imaginable that eventually, the US government owns the majority stake in these quasi-government-backed companies, moving the US economy further away from market-based capitalism?
Majority? That is a long long ways away. Top shareholder? It is true in Japan, so it is plausible. The Fed is not buying stocks at the moment.
https://asia.nikkei.com/Business/Markets/Bank-of-Japan-to-be...
The more concerning issue is that ETFs change management fees. Why is the Fed supporting management fees instead of buying the debt directly?