I'm not familiar with Stripe's situation, but there are non-public markets available for this kind of stock sale. You just can't buy from them unless you're already rich. I'd guess that long-term employees do have an amount of flexibility in that regard.
Can you form a mutual fund/ETF that invests into those kind of companies via the non-public markets and then sell shares publicly for the fund?
Publicly traded organizations can have any kind of private investment. I wonder, though, if there is regulation around how much of the public org’s capital can be put in private stock purchases…
I think some VCs are already also offering shares via mutual funds.
This is already a thing, large investors like Fidelity do exactly this.
e.g. Fidelity has a significant investment in SpaceX through a handful of their mutual funds, which you can then purchase and basically invest in SpaceX indirectly.
Unfortunately many companies have clauses in their options grants that prohibit employees from selling shares to any investor not approved by the company board (e.g. EquityZen).
I’ve seen companies doing buy backs to give people liquidity. In India, anyway. Check out Zerodha.