I wonder what things would look like today if the government had taken some portion of control over all the auto manufacturers, airlines, banks, etc it has bailed out over the years instead of just giving them unsecured loans

Would look like normal capitalism of the early XIX century, give or take. Bad, but not atrocious. Bailouts definitely wouldn't be abused as much, because, eh, they wouldn't be free.

And the old argument that public sector management is inefficient - well, it's not always a bad thing. It would then make sense for the government to re-privatize some of those shares, and use others for a source of income and a lever. And the companies bailed out this way would sink in power (which is good for competition), but not completely (which is good for their employees and economical stability). And, of course, I'll repeat about source of income. Perhaps there will be no more raising taxes with such a system in place. Perhaps even some taxes it'll be possible to simplify - any complex tax system works in favor of those who can afford to apply expertise, so those richer, and not poorer.

Also partial or full nationalization may sometimes work to good outcomes, while nationalized companies are less efficient, they also tend to retain institutional knowledge better, have more people working long on the same positions, follow labor regulations. For the telephone company or the train company or the central heating company or the public bus company it makes sense to be nationalized.

you raise interesting questions. how do you stop some asshole president from using infinite taxpayer money to manipulate the price of a stock?