The answer: of the 140 million tax returns filed in 2009, 90 million were filed by taxpayers that had adjusted gross income of less than $100,000 and that used the standard deduction. These taxpayers qualify for using the 1040A or1040EZ. So for more than 3/5 of US taxpayers, filing is not complicated at all.
Is the tax code too complicated for the other 50 million and for corporations? Almost certainly. But it is not a problem that afflicts the "vast majority" of Americans.
[update: according to this source, 32 percent of filers use 1040A or 1040EZ]