Your argument is well made, but alas I fear will not work. There are two reasons (among several) which I feel make it ineffective.
Firstly, there's an assumption that with economic schooling people would understand the benefits of delayed gratification and make decisions based on long-term outcomes.
I would suggest this is unlikely. We need look no further than health to see that people are very good at maximizing short term gain, at the expense of the long term. We know smoking is bad (long term) and yet it remains popular. We know exercise is good, yet it is clearly ignored by many, if not most. We are constantly educated around food choices, yet fast food and processed food dominate despite obvious long term effects. I would argue that economic education would be even less effective than health education.
We could further look at the levels of debt US folk are willing to accept to get short term gains. Some like housing and education offers a long term return. Others, like credit cards or vehicle financing simply trades short term gratification for long-term obligations. Again the prevalence of the long term outcomes does not appear to inhibit short term thinking.
Secondly, while the framing of your solution is a fit to US culture, the root of it is not.
You've suggested "shares in the economy" which "pay dividends". This lines up nicely with US capitalism terminology, but implies the shares can be sold. (And the ability to sell the shares undermines your argument, because once sold the under class is no better than before.)
So let's frame it differently, but also see how doing so makes it profoundly un-American.
Let's skip the "shares" part. Rather let the govt take your shares, and manage it for you. Dividends flow to you from birth. We could call this "basic income". You can't sell your shares (although nothing stops you selling your income.)
Since the govt would need ever more shares, the economy would need to grow larger than the birth rate (doable) and the govt would (over time) become the majority share holder in most companies. (Incidentally the Norwegian wealth fund is mostly invested -outside- Norway, which is useful, but would be impractical for the scale of the US.)
Personally I'm a fan of Basic Income. I believe that as production becomes more automated it is inevitable. But I accept that culturally the US can't do it - the idea of "money for nothing" goes against the very core of American culture.
Consider Covid as a test case. The US govt could afford generous support for citizens including cash. Supply chain woes, coupled with free cash, lead to some inflation. This inflation was lower than other countries. Supply chains recovered faster. Inflation dropped faster than elsewhere. Nevertheless citizens voted that administration out, largely because there was inflation at all. Instead they turned to a candidate who campaigned on raising tarifs (and all the education that the explicit goal of tarifs is to raise domestic prices seems to have been ignored, which is another nail in the economic education approach.)
Ultimately the administration which did offer "free money" (think also student loan debt relief) was punished at the polls for such un-American thinking.
So, let me be clear. I agree that having social support beyond your income is a good thing. Many countries already do it. But selling socialism is difficult in the US (not least because most Americans think socialism is the same as communism.)