Except that the government will just increase taxes on other common goods to make up for the shortfall.
So? Low income people still spend a lower percentage on their income on those "common goods" than the wealthy.
Actually, the opposite - low income people spend less dollars, but a larger portion of income, on those items than rich people. Plus, you really can't limit spending on "must-have" items like food, shelter, utilities, etc.
As OP stated, and I already repeated, a 20% tax on a $20,000/yr income is a much larger chunk of income than a 20% tax hit on a $200,000 income.
With a "flat tax," there isn't any way around that issue.
There are lots of ways around the issue.
You could tweak the system further.
Then it's not a "flat tax," it's a graduated tax system, like the one we already have.