So the system would still be plagued by tax cheats who hide income.
Yes, as with any system of taxation at all. However a flat tax set at 9% would have the least of that, far less than the current system. “Paying under the table’ for reason of tax avoidance might not exist at all because the reduction of dollar income that goes with that would simply not be worth it That flat tax should be the entire income tax that goes to the FedGov, including Social Security. Social Security is an income tax and there is no longer even a pretence that it goes into a separate set aside fund. It is general tax revenue and benefits are paid out of generalrevenues.