Neither customers nor employees can do that
In general, shareholder recourse is limited to electing a new board that will do what those shareholders want. The exceptions are cases where the actions of the corporation or board are discriminatory to some shareholders, or where there is fraud or self-dealing by insiders (e.g., shareholder derivative suits).
> They can also arbitrarily shuffle senior management as they see fit
Not directly. Shareholders elect the board, which then chooses the managers (CEO, etc.) who actually run the company. Shareholders usually only elect board members once per year at the annual meeting. So it’s not arbitrary.
> Neither customers nor employees can do that
This is true. Customers and employees don’t direct the corporation the same ways that the owners (shareholders) do. However, customers can boycott and employees can strike. Those can be effective ways to influence the direction of a corporation. Also, customers can sue for other things (e.g., fraud or malpractice), as can employees (e.g., employment discrimination). And any injured third party can sue the corporation.
I don't think that's actual true. As far as I know the only legal obligation a company has is to conduct lawful business.