Progressive taxes hurt poor people because they'll want to stay poor rather than having to pay more taxes.
But you are raising the floor. So poor jobs get better.
Transitioning is always problematic, but usually people will want to take better jobs, even for low payment increases, if they get them more prestige or decision space, or personal freedom. Also - there are many different jobs with different payment increase curves - so if thats (more money - I can 'feel') what you are interested in, there are those jobs as well. (Maybe less so than before.)
Usually the argument given is, that people would not want to get into the workforce if you dont put up 'existential pressure' (valid, but also mostly untrue..

). And not that people wouldnt want to grow on a job.
The American Dream is immortal.
The story? Sure. But what it stands for (if you work hard you can get a good life, because your earning 'upwards', in a growing economy) isnt active in american society anymore. Statistically. For most, it has changed to "The american dream of just getting by."
"Less valuable business models"? You mean like reasonable prices and good staff?
Viable. Dude has a business, hires another dude at low cost, scraps by. If that business doesnt produce 'living wage' for the other dude (lets say incl health insurance), if you raise minimum wage - those business either have to 'become better' or die out. In most cases, bosses cut a little less of the top to keep the business running, in some cases they will make the business leaner (but that leads to reliability problems), so eventually they end up innovating to get back on level (or get an even more viable business). And the ones that cant - die out.
But at the same time, if you've secured a job for years, and then you're fired because you're too expensive, what are you supposed to do?
Yes, looking at individual cases sucks. But remember - the idea is, that none of the promises of structural growth are working, service industry accounted for most of new jobs, automation is threatening even more good jobs down the road, and wage growth flatlined. So in the 'do nothing' model you end up with fewer and fewer people being employed, wages still not going up, everyone working taskrabbit and UBER jobs, and then falling out of the system, when a crisis like corona hits ("pseudo" self employed business models you chose between health and livelyhood).
So the idea is, that if you get fired, because you are working a business model that cant afford to pay higher minimum wage - you hopefully get picked up by investment projects that that tax money is supposed to go into. You get retrained. On the other side, you have to get retrained, because there are less 'low qualification' jobs around.
The people that benefit from high taxes need to get a better job.
There arent any around (in quantity).
Thats the key to all of this. Reps say - give tax brakes to companies, companies took tax breaks, invested money in stock buybacks, or in foreign ventures, but almost not at all in domestic production/innovation/wealth creation (wages).
In the past this (sometimes) did work. Currently it mostly does not, as projected growth potential is mostly seen outside of country, so thats where the investment goes.
How do you get investment back into domestic projects (not by doing PR stunts and giving grants to some companies, that will change 'voice' as soon as the grant runs out), but by funding new economies through state investment. Private sector had the chance (and will get it again). But currently they are down on investment potentials within the US.
Most of the jobs created since the financial crisis of 2008 (afair more than 2/3s) were low paying jobs in the service industry. Even if you work that job for 10 years, you are getting no where.
No, I'm only fifteen. The way to get out of poverty is to get a good job.
Unless you dont get education, got sick, got stuck in a service sector job and all 'good jobs' are creating projects in foreign countries (where the growth is).

This is not either or. You are correct also - I'm just showing you another scenario. Both are true at the same time. But again, the chances to get a 'good job' (stable, well paying, fullfilling, ..

) have become smaller.
The reason why it always works for the US is that people can get better jobs and not have to rely on tax-funded programs. My thoughts are we should raise taxes by 2% for ten years while slowly cutting back on government programs. Once those ten years are up, decrease taxes dramatically.
What do you do with the money then?
Three points.
- If you are raising taxes, you better should have a good plan, and a need for that money to go to. Otherwise I'm in your camp - don't raise taxes.
- Tax raises that are only time limited and then 'taken back' never work out that way.

If you got the money, you are spending it (as government, intelligently or not..

) to get back tax cuts, you'd have to vote in a different government, but thats ok. Thats fine. That works.
- Some of the changes we are talking about are structural changes (universal health insurance), so if something like this would get esablished, that means sustained higher taxes, but getting something for it in return (better health insurance, or a more pleasant more stable society (f.e.)).
If you can get away from the 'corporate always means more efficient' thinking, you should get it pretty quickly. If the corporate sector doesnt pick up the slack and performs (produces growth, or good stable (meaningful

) jobs, you pool in money in government, create new investment projects where needed, and if they take off you produce more market competition, forcing the corporate sector to perform (or pay better, or ....) and therefore innovate.
Again, tax rates were much higher in the US way back when - its just post war, when earnings from all around the world got syphoned back to the US producing a long lasting boom.
This 'ended' a little after 2001 (not double checked, please do), the slack was picked up by chinese money flowing into the US economy - people convinced themselves that it was creative financial planning that produced 'stable growth', then it became obvious that it was not and the bubble burst. There was no real sustained growth (house prices didn't rise as before). There hasnt been for quite a long time. (Look at medium wage growth curves, or GDP growth both with inflation factored out (GDP growth also has big company earnings factored in - so less 'shocking'

)).
The Idea on the dem side roughly is, to do health care system reform (you get scalped by corporations there), so you could grow a public health sector with little 'extra spent'. A little environmental tech spending (R&D), issue - that mostly produces growth long term if at all.
And again, if you raise minimum wage - you create innovation pressure in small and medium size enterprises as well. And you cut off the business models that only worked, by paying people below living wages.
People get more money, leisure spending increases > more business opportunities, ...
You just have to make sure, you dont increase income with people who would invest it in stocks, or in loan paybacks, because again - this doesnt create new investment opportunities ('growth potential'), thats just money that then mostly gets invested in india, mexico or china in the current economy. Again, you want to beak that somehow.
Big IF: State investment projects with tax money have to work. But the US has the health care sector it can reform (better for your entire society), and you have infrastructure projects that actually would need investing (moreso than europe afaik), ...
If they do not - you are correct and you just lost quite a few low paying jobs - and have higher welfare costs.
(Another reason why companies dont want to invest in the west currently is demographic development. In a society that first gets older, and then fewer, you have less 'naturally' occurring growth baked in. US is still rather healthy, but you are no baby boomer generation..

(India currently has one - so investment money goes there...

(Free trade agreements. (Which largely arent free trade agreements, but international legal guarantees for investor safety.))