That's just not true. The top 1% pay 33% of the tax (and the top 10%, "upper middle class", pay 70%). I think myth just causes confusion and misplaced anger.
Why does the relative amount of tax paid against the total networth matter in this instance?
Ideally taxes shouldn't be a large burden to anyone. The richer someone is, the larger absolute amount of taxes that person can pay for the same burden level. In fact, it's not just a fixed percent. The richer someone is, the larger percent of taxes that person can pay for the same burden level.
The rich got to where they are by using the infrastructure built by the government. A rich person gets more value from the government than a poor person, so thus it's fair that a rich person pays more in taxes.