They do not have an offer that will maximize shareholder value - they have an offer that will increase it.
Words have meaning and concepts have definitions. They are not arbitrary of flexible for the sake of making the argument you want.
The BOD's responsibility is to the company, a public company's responsibility is to the shareholders - this difference matters.
Cornell's legal information institute provides a nice and cited set of definitions (albeit in legalese) - https://www.law.cornell.edu/wex/fiduciary_duty
you have a company that is worth $1mil and we only imagine that it can be worth $100mil (based on the size of the market we are addressing). someone comes and offers $200mil but we know he will shut the company down (i.e. liquidate all its assets, or even simply the desire to destroy the company).
While this might be sad for the company, why is it not in the interests of the shareholders to take the offer?
so if Musk is willing to offer more than shareholders expect to see in the forseable future, why does it matter what will happen to the company after that? their interest in the company ends when their shares are purchased.
That is your assertion/opinion (yes, shared by many, sure). But it's not the only opinion in this case, and so I don't think the analogy holds.
Sure, I could see in the abstract times where an offer it just unavoidably good, and so it would not be in the best interest of shareholders to take it. But in this specific instance, there is a lot to be said on both sides of the offer (taking vs. rejecting it).
I would also argue that, depending on the purpose and goals of the company, knowing that a person intends to shut it down would be a reason to value existing (in order to continue carrying out their purpose) over money.
This isn't an argument for the poison pill clause, but is an answer to the following quoted question. It is funny to see this motivation of most existing shareholders ignored in order to bring into being analogous motivations of another. Especially when Musk's offer for Twitter has been in order to change it promote certain values, but conspicuously, better profits has not been one of those touted values.
> While this might be sad for the company, why is it not in the interests of the shareholders to take the offer?
I will sabotage my own argument somewhat though and say that I believe the economic motive dominates over time and is almost always (in macro and micro) the primary force.