"And if private contributions are outlawed, who pays for campaign costs? " Nobody.
Campaign spending should be rigorously limited to $20,000 for congress, $100,000 for the presidency. All funds must come from the candidate's own pocket. Abe Lincoln did not spend millions on his campaign for the presidency, but he won anyway. He did not have the power of the Internet at his disposal at essentially no cost either.
We need rigid laws forbidding all campaign contributions and limiting campaign spending to the amounts stipulated above. That is the ONLY way to get the power of government out of the hands of the criminals who call themselves "businessmen".