I am confused about your positions on preferred stocks - what you are saying does not track with what preferred stocks are today. Preferred stock only has a limited participation in profits, and common stock almost always outperforms preferred - for good casual reasons.
On to your points. There are models of what you are suggesting, most are predate industrialization (Railroads were to first to majorly break away from this model during the 1850, other companies followed) or are in Islamic countries (they still have traditional equity, but need the preferred stock to fill in the balance sheet since they can’t take out loans). These have had a poor record with larger corporations. Most of the problems boil down to having a permanent base of capital and liquidity.
A model might be partnerships or private bank shares. However, the more owners one has the more conflicts one has. Issues show up when you have more than 50 owners. Things break down when there are more than 250. You can reference any modern partnership (law firms, accounting firms, etc.) or worker owned co-op to see the numerous flavors of issues. Every year people are cashing out – they need the money, die, whatever. The demand for cash is often high, leaving scant fund for growing the business.
REITs and Limited Partnerships are another model. They issue and redeem equity frequently. However, since they don’t have a permanent equity and limited retained earnings, their projects tend to be of shorter terms and limited focus.
However, many industries need a long term capital base. Think fab plants, ship building, mining. These take years to build - companies can’t have their capital yanked out of them in the middle of these projects. Banks are special. Take a look at the financial crisis and the bank run on its equity. Governments still restrict how much capital they can pay out. Need cash to pay for an emergency? You may not be able to do that for years. These would be a hard fit in your model.
Another model you might look at are hedge funds and private equity. Like your suggestion they have a initial pay in. However many have lockout periods of 5 to 10 years, discounts on early redemption running from 20 to 40%. The reason for this tends to be liquidity (cash is often not at hand) but valuation is hard. You suggest that a person should be able to cash at a percentage of what a company is worth? Well, what is a company worth? Evaluating private companies is notoriously subjective. Differences of 20% between different appraisers are common.
A Dutch auction would solve some of the above issues, but then you are stuck waiting for management to offer the cash out. Need the cash at some other point in time? Tough luck.
Which takes me back to my original point. Investors in startups get paid when they cash out at the IPO. Anything that reduces the value of the stock on the secondary market is going to lower the price that I cash out as, which lowers my returns, which makes me less likely to invest.