Well, robert-c-, isn't this an interesting question. I'm going to confess that I hadn't until now learned of thematic criticism, which, if I understand my hastily undertaken research (the research was done in haste and I hastily dashed off to do it!), has at least some foundation in film criticism and is a reaction against contentless criticism found in modernist and formalist criticism. [Feel free to correct me where ever I'm going astray.] Thematic criticism also differs from the standard quest for theme by examining theme in relation to (1) the author's total life experience, e.g., the defining moment of inspiration, propensity for a given mind set, etc, and in relation to (2) a theme as it appears in the corpus of an author's work or as it appears in the collection of several authors works, thus uniting disparate authors by theme.
So--in relation to applying this to Shakespeare--hmmm--one point for which this might be interesting is to pursue thematic criticism across his body of work in relation to his own life to discern the influence of Protestantism in his life and in the social structure(s) of his day. The example that comes most immediately to mind is Hamlet's indecision as a direct result of his Protestant beliefs in conflict with his father's religious beliefs, which embrace revenge whereas Hamlet's don't. Another idea that springs to mind is to apply thematic criticism to the collected works of Spenser, Sidney, Marlowe, Jonson and Shakespeare to discern their various stances on the mimetic ideal given voice to by Sidney. Perhaps one might determine where the mimetic notion began to break down, as break down it surely did.
So--from these random musings I think I can suggest that while there may be some objections that I don't yet recognize to applying this criticism to Shakespeare--as I don't yet know enough about Thematic Criticism to have objections--I can see how the application of thematic criticism to Shakespeare might yield some useful and enlightening results. There is one text from Harvard English Studies saying of itself that it is the best that looks interesting: The Return of Thematic Criticism, essays edited by Werner Sollors.
Here are a few links for others who, along with me, may not have encountered thematic criticism before:
Thematic Literary Criticism
Littérature et Sensation by Jean-Pierre Richard
Thematic Criticsm Method (in films)