The capped-profit / non-profit structure muddles that a little bit, but the reality is that entity can't survive without the funding that goes into the for-profit piece
And if current investors + would-be investors threaten to walk away, what can the board really do? They have no leverage.
Sounds like they really didn't "play the tape forward" and think this through...
But until he is re-hired Sam Altman is to all intents and purposes fired. And it may well come to that (and that would almost certainly require all those board members who voted for his ouster to vacate their positions because their little coup plan backfired and nobody is going to take the risk of that happening again, especially not in this way).
Boards are agents to their principals. They call the shots only as long as their principals deem them to be calling them correctly. If they don't, they get replaced. Said differently, board members are "appointed" to do the bidding of someone else. They have no inherent power. Therefore, they do not, ultimately, call the final shots. Owners do. Like I said, this situation is a little muddier because it's a non-profit that owns a for-profit company, so there's an added layer of complexity between agents and principals.
OpenAI isn't worth $90B because of its non-profit. The for-profit piece is what matters to investors, and those investors are paying the bills. Sure, the non-profit board can fire Altman and carry on with their mission, but then everyone who is there "for profit" can also pack up their things and start OpenAI 2.0 where they no longer need the non-profit, and investors will follow them. I assume that's an undesirable outcome for the board as I suspect the amount of money raised at the for-profit level dwarfs the amount donated to the non-profit... which effectively means the for-profit shareholders own the company. Hence my original comment.
Yes, they are accountable (and I'm actually surprised at how many people seem to believe that they are not), but they are not without power. Legal and practical are not always exactly overlapping and even if the board may not ultimately hold practical power (even if they believe they do) legally speaking they do and executives function at the pleasure of the board. If the board holds a vote and the bylaws of the company allow for it and the vote passes according to those bylaws then that's that. That's one good reason to pack the board of your billions of dollars worth company with seasoned people because otherwise stuff like this may happen.
Afterwards you can do a lot about it, you can contest the vote, you can fight it in court, you can pressure board members to step down and you can sue for damage to the company based on the decision. But the board has still made a decision that is in principle a done deal. They can reverse their decision, they can yield to outside pressure and they can be overruled by a court. But you can't pretend it didn't happen and you can't ignore it.