In response to a previous question Sean Greenberg characterized philosophy as
Perhaps Professor Greenberg should reply to this, but here goes: I suggest that there are at least two ways of defining a philosophy. On one meaning, to have a philosophy is to have a worldview or a conception of yourself, the world, values, and so on. From this point of view, most people have a philosophy Secondly, "philosophy" can stand for the disciplined reflection on world views or ways of thinking about reality and values. The latter can certainly involve description, clarification, and criticism. Probably Professor Greenberg put such an emphasis on arguments is that while philosophy can involve a great deal of exploration and exposition, a great deal of philosophy addresses questions of justification or evidence. Using these distinctions, I think it likely that Shakespeare the person had a worldview and thus had a philosophy, but in the work attributed to Shakespeare there are multiple philosophies or worldview (Macbeth's philosophy seems different from Prospero's) and it would be hard (but not impossible!) to find straightforward philosophical arguments in the texts that would help us choose which philosophy is better justified.
To speak to your final suggestions, I do think that philosophy need not be seen as so defined by argumentation that this definition becomes a straitjacket. After all, the term "philosophy" come from the Greek philo and sophia and is usually translated the love of wisdom. So, in a sense, loving wisdom can be a philosophical activity, and perhaps a wise person is not always argumentative! As for philosophical work on literature and the arts in general, check out the online site for the American Society for Aesthetics and the British Society for Aesthetics.