Imagine your employee base graphed on two orthogonal dimensions: their performance level, and their motivation for working for you. (In reality, "motivation for working for you" is actually many dimensions, and performance is likely to be somewhat correlated with certain motivations, but this illustrates the point at least.) The buyout offer doesn't affect people who aren't financially motivated, so take them out of the equation, and look only at the folks who are strictly financially rational.
You can loosely group them into high-performers (those who are known achievers, who always have an offer in hand as soon as they make overtures toward searching), mid-performers (those who pretty much amble along, and for whom the job search provokes intense anxiety, because they typically have to put in 3-6 months of full-time searching to find a new position), and low-performers (those who don't do anything and everyone already knows it). In most companies, the distribution of these is roughly 10-80-10. You can fire all the low performers; the point of bucketing them into the "low performer" category is that their underperformance is obvious enough that they leave a documented paper trail that makes this unambiguous. And in a company that's run into financial difficulties, the top 10% has probably already left, because they continually get job offers and when your company is going down the tubes, it's to their advantage to take them. That means that most of your financially-motivated employee base probably fits into the middle range, the folks who are average employees and will have average difficulty finding a new job.
The point of a buy-out is to reduce the anxiety levels surrounding getting that new job to the point where anyone who is unhappy at your company but isn't sure they have enough money or skills to go elsewhere ends up taking the leap.
Unless you're not confident enough that you'd be able to get a job in those X months, either because of your skills or there not being jobs around.
Also, even if you like your coworkers, or want to acquire skills, if the ship is going down and you feel confident enough that you'd be getting another job quickly, then you might jump ship anyway.
What a buy-out does is take money out of the equation. It makes it so that everyone who is there for financial reasons has a financial reason to leave. This can be logical if the company has little to offer financially, as in, for example, a company that's in bad shape during the financial crisis. The only employees who're left are those who are not in it for financial reasons - the true believers in the company, those who like their coworkers, and those who want to gain skills. Those people will be a lot happier when they don't have to listen about their financially-motivated colleagues grousing about how the company is going under, and they'll be more productive when they're happier.
The CEO's fatal mistake was that he forgot that his board is financially motivated. From their perspective, he's acting insane, because why would anyone stay at a company for reasons other than money? And they're his bosses, so they can have him removed. For this to have worked, he probably should've taken the company private, or at least packed the board with people who bought into his vision for what kind of company it should be.
The board's fatal mistake was in removing the CEO. By offering buyouts, he had just aligned the company in one direction. By removing him, they suddenly took the company in a different direction - one that they had less than zero capital in, because the CEO's previous actions had alienated everyone who might've helped them.
In both cases, the failure was in understanding that people are different and may have different motivations, and that a company survives based on how well it can hold together people who all have their own goals and worldviews and motivations.