My understanding is that while yes, in some sense a flat tax is neither progressive nor regressive, that's not really what people are talking about when they say a flat tax is regressive. Instead, the issue is that the poorer you are, the larger percent of your income is spent on essentials.
If I go from spending $200/month on groceries and then I get an enormous 300% raise at work, I'm not going to start spending $800/month on groceries, maybe $300. The rest goes into the discretionary part of my budget. So, to someone making $40,000/year, paying 20% of that in rent, 10% for food, 20% for healthcare, and 15% for other stuff, adding a 15% tax is a decent chunk of the rest of their budget. But, for someone making $400,000/year, paying 15% for rent, 2% for food, 10% for healthcare, and 15% for everything else, then 15% of their income towards taxes is comparatively much less of their total budget!
This is true, even though everyone is paying a proportional amount of their income. Another way of thinking about this is as follows: if there is a homeless woman on the side of the street who makes $30 one day begging for change, should she be taxed at 15% and have to give the government $4.50 because it's "fair", or, should she be receiving more value in benefits than the average citizen because there is a floor on how much you can spend on food and survive?
Thus, even a flat tax can in practice be regressive, because essential spending does not scale with income.
An addendum: While it is not relevant to this argument, I also agree with the other commenters that whether a tax system is flat or has marginal brackets is not the main source of complexity, and is relatively super simple. The main complexity comes from determining which sources count as income. Presumably, birthday presents don't, but whoops, now there's an exception. And this is suddenly how tax laws get super complicated.