The intention is that the government won't maintain control of any of the companies it took a controlling interest in, but sell the shares back once stock prices rise to recoup it's investment.
That was covered, but perhaps you missed it.
Secondly, I can't think of a major privately held car company - even Venture is publicly traded.
Finally, if there were any privately held car companies, they still have the same advantages they ever held over monolithic corporations - agility. They should, in theory, be better able to adapt to consumer desires than GM did, for example.
When consumers were looking for fuel efficient cars, GM brought back the Camaro (neat car don't get me wrong), and bragged about 20mpg! Little off the mark.