Even if you did this you would still need to pay yourself a salary or dividends, either of which would then be taxed as income. So there's some wiggle room to avoid paying tax on business expenses, and I'm sure that would be abused to an extent, but given that the tax system would likely still be progressive there's on reason the poor would be paying a larger share of the tax.
Under current law, if you live rent free in a house provided by a corporation that you owned, you anyway have to recognize an appropriate amount of personal income for that. Same with the use of a vehicle.
Compliance in such situations would probably be lower with no corporate tax, by incorporating isn't some magic process that makes income disappear.