Hey there! Thank you so much for your comment, and congratulations to both of you for engaging in dissent!
I'll answer the hard question first, and then make an observation to your friend about their views.
The reason a flat tax is immoral is that your first dollars covers your absolute NEEDS, and with each additional dollar, it gets more and more into "wants". If someone has just $10 a day to live off of, taking $3 of that is literally taking food out of their mouth. Never mind their lack of shelter, or any other of their basic needs. If someone lives off $100 a day, taking $30 leaves them with $70; not only is that enough to feed them, but likely get them basic shelter. In a just world, zero dollars of the first person's $10 should be taken - he needs every dollar he can get - the difference being paid by those who won't lose a meal by contributing a little more.
Put simply in this example, a flat tax leaving each respective person $7 and $70 is far less just to their outcome than leaving them with $10 and $67; that missing $3 hardly effects the second person, and makes a huge difference for the first person. Your first dollars are oh so valuable.
This is why many on the left would support a 0% tax for your first $30K; you don't make enough to really pay a meaningful amount of taxes, and those dollars will go a long way. That being made up by a 1% tax on millionaires, which is something they would hardly even notice.
But I think at the heart of your friends point is not that she advocates for a flat tax - her own comments saying as much - but that she wishes the tax system was simplified, and that is very much something we should all agree on.
When she says she would like a flat tax that changes based upon income level, she has just suggested a progressive tax system, the alternative to the flat tax. And that's what we have now, tax rate based upon bracket.
But the tax system is limitlessly complicated - and that's by design. A savvy businessman (or a corrupt one, depending your your perspective) can use tax law to avoid paying taxes. It sounds like your friend is for a system that says "No, this is your income, this is your tax rate, the end." And that's absolutely a great position to stand on - I'm with her - but that's not a flat tax, that's a simplified progressive tax system that does away with all of the deductions and black magic we go through year after year.
A flat tax means "everyone pays the same [flat] level". I hope I've demonstrated why that's unusually cruel to the poor, them parting ways with valuable dollars so the rich can keep a few more "nicer trim on my fourth car" dollars. It sounds like your friend understands that, and thus supports a progressive tax, not a flat tax.
But she, like many, are exasperated by the wealthy using our intentionally complicated tax system to avoid paying anything at all, while the rest of us get stuck with the bill. She's dead on.
A progressive tax system to not punish the poor. A simplified tax system to ensure the rich can't get off the hook.
Hope this helps!