They don't actually try to pass a flat tax because it is not feasible to actually do a flat tax in a country like they US with such a wide range of incomes. Even with a low rate, there will be people poor enough that paying the tax is a major hardship. So every even remotely serious "flat" tax proposal has some cutoff and only applies to income above that cutoff.
But then it is not actually a flat tax. It is a progressive tax with two brackets. Once you get there, it is really hard to come up with a convincing argument that two brackets is better than more brackets.
If the calculation wouldn't mystify most taxpayers, what would probably be the most sound theoretically would be a continuous bracket structure.
The other reason they don't do it is that the flatness of the rate structure has pretty much nothing whatsoever to do with the complexity of a tax return. If you changed the rate structure from a progressive rate with several brackets to an actual flat rate, or a more realistic two bracket progressive system, that would shave about 1/4 of a page off the several thousand pages of the tax code and tax regulations.
The complexity of it is all in figuring out which income is taxable. Once you've got all the income classified into the various relevant tax categories, the actual calculations where you use the rate structure is trivial.
With a tiered tax system, individuals need to be the ones who pay taxes because people who combine income from multiple sources need to pay differing amounts of taxes. With a flat tax system, all income is treated exactly the same so governments can switch to collecting income taxes from corporations. Corporations basically report the amount they spend on payroll every year and pay some percentage of that as a tax to the government. Then the government takes some portion of that money and remits it back to individuals as a negative income tax (essentially a UBI).
Flat tax changes the way money flows, instead of company -> individual -> government, it goes company -> government -> individual. It's both easier to tax corporations than individuals and easier to give money to citizens than take money from citizens. These two powerful benefits alone are meant to make up for the severe rigidity and lack of flexibility that flat tax systems intrinsically have. It's an open question whether these two things are enough to make up for all the flat tax flaws but I don't think there's an obvious right or wrong answer that we've discovered yet.
Also what about income from investment?
Income from non-salary sources aren't generally covered by flat tax proposals because they're so complex and the existing tax system would still exist to deal with them. However, for the vast majority of citizens, salary is their only source of income and their experience with the government becomes "how much can I convince the government to give to me" rather than "how do I try to avoid the government from taking from me".