Never questioning the value of philosophy is unphilosophical

Philosophers are supposed to be asking fundamental questions and examining background assumptions that are usually taken for granted. Why do most philosophers then avoid asking inconvenient questions about philosophy itself? Stefan Schubert finds the general lack of metaphilosophical reflection unfortunate:

"What is the role of philosophy - a discipline that inherently has a rationalistic bent - in an academia dominated by empirical ideals? ... After the demise of logical positivism and linguistic philosophy, analytic philosophy has unfortunately mostly avoided metaphilosophical questions of this kind. ... [M]ost analytic philosophers still seem happy to carry on discussing small bite-sized problems without worrying about methodological questions. This in my view is most unphilosophical." (Schubert: Ernest Gellner's Words and Things: A Case Study of Empirical Philosophy, 2015)

Philosophers have never been afraid to attack sacred cows and common sense or take extreme positions on issues. For example, they have denied the existence of qualia, beliefs, desires, tables, baseballs, free will, objective moral facts, knowledge of the external world, and so on. Peter Unger even denied the existence of himself! But when it comes to metaphilosophy, questioning the value of philosophy as a whole is suddenly out of bounds. At best, philosophy is just assumed to have some value.

Philosophers have of course criticized other subfields or movements of philosophy, such as metaphysics or phenomenology. But I'm talking about across-the-board anti-philosophy. Not even naturalists oppose philosophy as such, just the parts that are too far removed from science. Similarly, Wittgensteinians who think that traditional philosophical questions are just pseudo-questions still leave room for their own brand of philosophy.

This is puzzling and cries out for an explanation. After all, radical, all-encompassing anti-philosophy is just another metaphilosophical option on the table, and it follows quite naturally (pun intended) from already popular positions such as naturalism. It is not obviously incoherent either (this and this). But I will leave explaining this glaring metaphilosophical blind spot for another day.