After writing this, I realized that that is kind of how businesses used to work. You would take a few rounds of funding from friends, family, and eventually connections/PE firms before going public. The whole idea of IPO-ing after series F is the new, weird thing.
The best way out once you're in this deep is probably an IPO. The investors get their exit, and you're now "only" beholden to shareholders, who are more used to sustainable businesses that can survive decades or centuries.