The main argument for a flat tax is to simplify the system and make it harder to game. What really punishes the poor are complicated taxes--the kind that people who can afford lawyers and accounts don't pay.
Singapore and Hong Kong seem to do the right thing in terms of taxation: their systems are simple and the income tax is progressive, but the top marginal rates are pretty low by European standards (around 20%).
To help the poor, universal basic income seems like a good idea. It pairs well with my favourite kind of taxation: land value tax. See https://medium.com/basic-income/why-land-value-tax-and-unive... for a pretty insightful rant.
If an income tax is necessary at all, giving everybody a flat allowance per head plus a pretty flat rate should be good enough on the poor? (Otherwise, do whatever Singapore or Hong Kong or Estonia are doing.)
Even in your follow up post as soon as we make provisions for the reality we live in the flat tax goes back to being a progressive tax, again.
So what we really need is a simpler progressive tax rather than a gradually complex flat tax.