Seems to me GM and the banks need the same thing: people who have sufficient income to buy cars and make mortgage payments.
In general, direct public spending is for public goods: military, legal system, various infrastructure, etc.
To support private sector output the way to go is to support demand, and let consumers decide which products succeed and which don’t.
If government had declared a payroll tax holiday (treasury makes the fica payments for employees and for the business) three months ago, car sales would be up and mortgage delinquencies down.
And don’t forget to toss in something that reduces fuel consumption sustain our real terms of trade.