If everyone just pays a flat sales tax rate, the poor bear most of the economic burden.
No. If everyone pays a flat sales tax rate, the people who spend more will bear most of the economic burden. The poor would pay a smaller part of the overall economic burden (because the poor spend less money in the overall economy than the rich do.)
It is possible and likely that the burden on the poor would be more as the percentage of they money that they have that goes to taxes could be higher, but it is inaccurate to say that poor would pay more of the overall burden to the economy a.k.a. "the economic burden".
Please do not confuse "the economic burden" with "the burden on the poor". It is an important distinction. They are both important issues, but they are different issues, and should have different arguments and conversations behind them.