You've made a lot of assumptions about what a flat tax implies.
You can do much like Trump did with deductions and define a significant standard deduction, and just have that with a flat tax. Yeah you don't make much, then 10% may hurt, but 10% also isn't large and thus the deduction is more likely to cancel out your tax burden the less money you make.
For grins, lets just say that it is the same as 2019 and we leave out the head of household math and ignore kids and dependents. We'll go with the single renter for now. So, $12,200.
Lets say you have a flat tax of 18%.
Someone making $100,000 a year would pay $5,800 in taxes. Anyone making $67,777 or less would pay nothing or get money back. Theoretically someone making $10 million a year would owe $1,787,800. But for that to be true, you would not have to differentiate between capital gains and income.
The problem with a flat tax isn't dealing with the low income people fairly. Most flat tax proposals deal with it one way or another. Either via a deductible, refundable deductible/credit, or by exempting the first X dollars of income. It's that the task of making a simple income tax system is not possible. It either needs to be complex or touch non-income taxes rather thoroughly.
An example. Poof, capital gains is income. OK. I'm rich. I just form a holding company, own THAT, and it owns the stocks and investments. It can sell without me realizing a capital gain. Now you have to make it complex, or extend the flat tax idea to businesses.
It actually much easier to keep it form being regressive than something like a VAT. VAT is how most "free" healthcare is paid for. Warren buffet may make a bajillion times more than his poor secretary he always talks about but he's never just going to buy enough shit to make a 17% vat place a meaningful tax burden on him relative to a family of four making $52k a year.
Which leads to the thing lots of people like ignoring (or just not realizing), which is capital gains is basically VAT, but only for certain things.
Most discussions of tax reform are naive because they ignore the fact that you are talking about how to fund a government, Which essentially means you are talking about changing how that government works.
We've gotten to a place that is overly complex and messed up. Most people seem to think something needs ot be changed. what I can guarantee is nothing that happens will be simple. It can't be unless your government is simple. And frankly we expect too much off it for a simple system to work. Do you expect roads and a military? Yeah you already passed simple.
However, we are at a point where shit has just been glued on that continuing to do so means that broad swath of citizens are put in financial peril every time someone wants more tax revenue. There is a good bit of hostility towards social programs because the anti-pattern we have seen is that poor people have a problem. Everyone is sympathetic to it, then a bunch of millionaires decide that the middle class needs to pay for it. And even if you have a good idea better than the last one, we never seem to dismantle the old money pit to try a new better idea. The middle class has been dealt with in bad faith for decades upon decades. Saying we will fix it by having the millionaires tax the millionaires is just stupid naivete. We've been down that road. They will jsut see it as an opportunity to manipulate voters by threatening that the other guy will take the new thing away and will make a program and shift the cost of it to the middle class. Then you have people who are like fuck it, I'll keep my money, the rich and poor can go die in a fire. Some others are also naive and believe that if they just make the tax say 20%, the goverment will get their fiscal shit together and deal honestly with the middle class.