The early years Amazon wasn't profitable by choice. They could have stopped that at any time and even did demonstrate it by having a single quarter with $1 profit or some such.

Anthropic and OpenAI have no choice but to go public if they want to avoid bankruptcy. Venture capital firms are struggling to raise more capital, the bond market is so saturated that the borrowing costs are getting too high and big tech is also at the limit of how much they can invest, all while AI companies' costs are going through the roof. Retail investors is the last market they haven't tapped into and to do that, they have to go public. There's just no way around it.