> I wonder: if you added it all up, would a flat tax (which is nominally regressive) actually be more progressive than the regressive taxes we have?

Absolutely not. The US has the most progressive federal tax code in the OECD, mainly because we don't have a VAT like most other countries.

Nearly all of the loopholes you mention are at the federal level, where half of the households in the nation pay <= $0 in income tax.