...I suppose it'd be simpler to say that a flat tax has a disproportionate impact on lower incomes.

I guess there are two issues: (a) the definition of progressive/regressive and (b) the justification for progressive taxes.

If the taxes paid by an income group as a percentage of that income rises with the level of income the tax is progressive. If this falls as income rises the tax is regressive. Of course the categorization depends on a calculation of who actually pays a tax and economists have their own way of doing this.

The justification for progressive taxes is not usually based on fairness. Economists find this to be a slippery concept. The argument is utilitarian: Progressive taxes produce less pain for a given amount of revenue raised. The idea is that the reduction of utility (pain) caused by depriving someone whose income is $200,000 of $50,000 worth of consumption is less than the reduction of utility caused by depriving someone whose income is $40,000 of $5,000 worth of consumption.

Said in reverse, going from 50K to 45K has a larger negative impact than going from 250K to 245K. Maybe 50-48 utility lost equals 250-245 utility lost (numbers I picked at random) so it maximizes utility to tax the higher earner more on the margin, no?

Is a VAT considered regressive, progressive, or neutral? I honestly don't know, but my gut says "neutral" since it's based on consumption, not income. What's the expert view?

spend a larger percentage of their incomes on taxable goods

Better concepts include a refund or "pre-fund" of the tax value of a given standard of living, offsetting the basket of food, fuel, etc.

People with low incomes save less. That's the basis for the regressivity.

Yes. I guess you could wed a negative income tax to a VAT and claim that the combination is less regressive or not regressive but the VAT part is still the VAT part.

the concept from the mid-90s under the "Fair Tax" label did not have a 'non-taxable goods' category. If you bought something, the transaction was taxed. If you paid for a service, the transaction was taxed.

Monthly Universal Basic income voucher pre-pays the cost of basic goods and services, so low income has a net zero tax burden at worst, net tax income possible.

Prompted me to go online (because the answer surprised me), and I see your answer stated verbatim in a Tax Policy Center piece on this question. I hadn't though it through and was curious about the data that lay behind the answer, but the TPC answer made it quite obvious without the data.