This is exactly right and it is intentional. This is just like withholding taxes that hide from people the true cost of their tax burden.
The power-hungry politicians WANT to keep people ignorant of the true tax burden. Complaining about "corporate taxes" just confuses the ignorant and uninformed which serves only politicians. It is useful to them also to divide and conquer so that there are fights just like this. Where one side is arguing for higher corporate taxes "for fairness" while not understanding that THEY and everyone else are going to be the ones paying.
It is a great ploy from Washington (and other world capitals) to keep the un-educated upset and ready to follow them over a cliff.
The best solution is as you said - people pay taxes, and there is no withholding. There is no masking of the size of government and the tax burden. No hiding behind ignorance while allowing yourself to be manipulated.