Let me try to explain with a very very simple example why the average person might pay less, even though the average tax might be higher.

Person 1-10 make $10. Person 11 makes $100

Case 1, a flat tax of 30%

Person 1-10 pay $3, person 11 pays $30, for a total tax of $60 on $200 total income, that is 30%.

Case 2, a progressive tax of 1% on income up to $10, and a tax of 80% above that.

Person 1-10 pay $1, Person 11 pays 1% on $10 and pays 80% on $90 which is $72, a total tax of $82 on $200 income, a 41% tax!!!