I think what we actually want is the ability for us normal people to invest in private companies more easily, rather than for private companies to become public more easily. This is difficult partially due to the SEC and partially due to historic reasons for how private equity is generally allocated and traded, but I'm hoping it gets better some time.
But I share the sentiment, I'm continually disappointed when there's a company I really want to invest in, but I know that I won't get a chance to buy it until 5 years later when it has already 100x'd and then just gets IPO dumped to retail, becoming more of a short-term casino (e.g. see all recent tech IPOs) instead of long-term bet placed intelligently early-on in the race.
If I could purchase shares in anything from Stripe to Discord to even Chik-fil-A (not to mention a certain 10 or so certain YC companies...), I would in a heartbeat (and I'd be willing to pay a premium for this too!), but the privilege likely won't be given to me until I no longer desire it.