The natural reaction should just be to stop taxing citizens on worldwide income.

One reason that could explain the difference between corporate and private taxes is that the companies can (in theory) prove how and where went/came from.

It would give a huge advantage to foreign companies. It would strongly discourage investment in US companies.

Please read my comment again. It would apply to every corporation doing business in the US. Regardless of where they're headquartered. And any corp of significance has to do business in the US - it's the largest market in the world.

Then wouldn’t other countries immediately retaliate with the exact same tax, but applied to American companies doing business in those countries?

True. But they'd also have to raise their tax rates to American rates, for the taxes to have any effect.