I would assume you would also like to tax foreign investors. Corporate taxes allow for this thus lowering your and every other US citizens taxes. Which is why there a 'good idea' and not just a meaningless exercise.
That's a good point. On the other hand, 0 corporate taxes would lead to more investment by foreigners, and some of that investment would make its way into the local economy.
As to which effect is stronger, I have no idea!