Earlier quoted context omitted.
The regressive aspect of it is offset by the large standard deduction.
That still won't solve the issue. The reason for progressive taxation is that there simply isn't a tax rate that allows you to both sufficiently fund the government and doesn't punish low income earners. The gulf between what the lowest and highest earners make is simply so vast that you can't do it without progressive taxation.
Sure there is. After eliminating deductions and special treatment (including, e.g., for capital gains), do a flat income tax at the current maximum marginal rate with a standard exemption at the level that results in the same median total tax rate as currently.
You'll more than make up on the high end the revenue you’ll lose on the low end. Of course, all actual flat tax proposals are designed to be massive tax cuts, especially for the rich, which is why they tend to feature a flat rate that's similar to the marginal rate for one of the lower brackets. Do that and you can't pay for much government.