Milton Friedman used to say that a flat tax of 15% would be enough to replace the entire scheme of income tax. There is also one major boon: lots of capital is spent on taxes, from accountants to lawyers to gubernamental organizations, etc etc.
An effective 15% rate without exemptions would fly very well with the right (simple flat taxes) an the left (no exemptions) but as MF said, the political stances on this topic dont trust each other and they are both right: even if we moved to such a system, the right would still advocate for exemptions and the left for increases in taxation.
Its a noticeable deadlock.
I also want to mention that flat taxes are not regressive by definition. The progressiveness or regressiveness in that case requires to see where the taxes are spent.