Microsoft's relationship with OpenAI was really ideal from a speed-of-advancement perspective. That is, reams and reams have been written about how Google has to move at such a slow pace with AI productization because, essentially, they have so much to lose. Microsoft saw this first hand with their infamous Tay AI bot, which turned into a racist Hitler lover in a day.
Microsoft's relationship with OpenAI was perfect - they could realistically be seen as separate entities, and they let OpenAI take all the risk of misaligned AI, and then only pull in AI into their core services as they were comfortable. Google's lack of this sort of relationship is a direct hindrance to their speed in AI advancement and productization. Microsoft's lack of a board seat gives them a degree of "plausible deniability" if you will.
Plus, it's not like Microsoft's lack of a board seat impacts their influence that much. Basically everyone believes that the push to get Altman back has Microsoft's/Nadella's fingerprints all over it. Their billions give them plenty of leverage, and my bet going forward is that even if they don't take a board seat outright, they will demand that board membership be composed of more "professional", higher caliber board members that will likely align with them anyway.
Read a good a article about the history of the OpenAI board that argued this all went down due to the recent loss of 3 board members, bringing total board membership from 9 to 6 (including losses like Reid Hoffman, who never would have voted for something like this), and Altman wanted to increase board membership again. Likely the "Gang of Four" here saw this as their slim window to change the direction of OpenAI.
It doesn't. It never does. Especially when you're not profitable.
What I'm fairly sure of though is that if the board had been stocked with heavyweights rather than lightweights that this would have been handled in a procedural correct way with a lot more chance that it would stick.