What is the importance of value maximization in the theory of the firm?
It would be difficult to overstate the importance of value maximization in the neoclassical theory of the firm. Value maximization is essentially the core of neoclassical firm theory. It says that all firms do — or should (it's often vague on precisely that point) — seek to maximize their total market value, and maximizing social welfare as a result. There is a mathematical theorem that supports this, which rests upon extremely strong assumptions about complete markets, perfect information, and perfect rationality.
Of course, none of these assumptions are anywhere close to true, and if you relax them even slightly the whole theorem falls apart (as shown in the first linked paper), but this has not stopped value maximization from remaining the core of neoclassical firm theory. Its chief competition is stakeholder theory, which essentially argues that social welfare would be maximized if firms sought to maximize social welfare. Stakeholder theory has often been argued to be trivial and useless (as is argued in the second linked paper), but, unlike value maximization, it at least has the virtue of not being obviously and completely wrong.
At best, value maximization might be a useful descriptive theory of how firms do behave, and perhaps there is some way of showing approximate value maximization is approximately social-welfare optimizing under more realistic conditions, but surprisingly few economists have even attempted such an argument. Instead, there is this weird notion that if the math works in some idealized imaginary universe, this must actually be a good policy in the real world.