I think you’d be stretching facts if you say that most people support regressive (or progressive taxation as you call it).
Most people don’t know too much about taxes(me included) and just bear with them.
I’m genuinely surprised that people believe that regressive taxation is better than a flat tax. It’s only the uber rich who are generally most aware of the tax policy and often lobby for and dictate how it’s written. And across the board, they support regressive taxation.
The main complaint of the existing system is that the rich don’t pay their fair share and the middle classes are taxed at a disproportionate rate to keep the rich insulated.
The issue about taxing the poor is non existent, you can’t tax someone who doesn’t have any money.
I’m all for basic services for all and for a fair taxation policy, sadly I don’t believe that the regressive policy meets those goals.