You're asking several questions here.
Lets back up a bit and consider what non-dividend stock is, why it might exist. After all, we can't decide if its foolish or fraud, until we see who might buy it, and why.
So, a non dividend stock is one that habitually retains earnings (profits) or absorbs losses, in the hope of stronger future growth. Literally, dont give away adollar today, because we can use it to make $1.50 tomorrow... or $5 next year.... or $40 in 30 years... or avoid future risk, or provide resources to fuel future profit streams/projects, or undertake R&D to protect our market position, or expand, or.... or.....
With that understanding, we can easily see why someone would invest. They feel the future prospects are better with that dollar retained, than with it paid out now. Or, more generally, they feel the company will be worth more in future if it habitually retains its cash than pays it out to investors.
If Im an investor, with $1 million in Amazon (or whatever other company you have in mind), perhaps I have a choice. They made profits equivalent to $70,000 on my $1m value investment. I can either be paid that $70k, or ask them to hold onto it and use it in their future business. If i can only invest that $70k at 3% and I believe Amazon profits can grow at 5% if they hang onto their earnings instead of paying them out, then logically I ask Amazon to do just that. If I want the certainty of cash in pocket, i dont invest in a company that believes in total reinvestment of earnings, because of the conflict over dividend policy.
So, your questions....
I'm trying to see why a non-dividend stock isn't subject to the Greater Fool Theory. There's no intrinsic connection between the stock price and actual company profits without dividends.
Oh but there is. The connection is because stocks arent priced based on distributed earnings. They are priced based on sentiment and future value perceptions. That future value can be measured in terms of profitability and market position/prospects and culture and management competence and tax regime changes and a thousand other factors, whether or not the earnings are paid out.
Investors constantly estimate what a company may do, in terms of growth or change, and risks faced, in the future. What it pays out or doesn't, doesn't much change that exercise.
Now, we do assume that ultimately the value will be realised/realisable. That is, at some point somehow, Amazon (or whoever) will have a market value of $X per share, built up in part by resources created by all that retained earning, and we will see that materialise as cash.
But whether it does so in a year, a decade, or a century, that value is real. Unlike a ponzi scheme where it never was real in the first place, it was only a fake, a charade.
So my question is... suppose a stock price were to continuously drop despite high company performance (profits)... Is there something the shareholders can do to get some share of the profits? Only thing I see is voting for the board of directors in the hopes new members will put in a dividend plan? Is there anything else?
If the stock price is constantly dropping despite good (growing) earnings, then theres a problem. You have to ask why. Perhaps its in a sector falling out of favour (high street stores when internet channels got big). Perhaps the market doesnt like how its run. Perhaps its seen as not managing itself well, or vulnerable in some way to some serious business risk or upset.
As the market constantly re-evaluates the company, it may decide tomorrow, the business is worth a bit less than today (or not an enticing amount more). That may deter some investors, and lower the price.
If that continues, eventually there will be investors who hung on to it unduly, even while "the writing was on the wall", and rightly lose their money for being fools. There will be other investors who thought it might grow, took a calculated risk, lost some cash, re evaluated, and sold it on, based on a price that seemed fair on the day (to the market).
But at any stage, if enough investors want a dividend, they ultimately control the directors (by appointment/removal), and thus can make it so. But as they all invested in a company with a non-dividend policy, perhaps the one thing they may all agree on, is they dont want that....