Shakespeare's religious ideas are hard to pin down, and are compounded
depending on who you think "Shakespeare" really is! It's safe to say that
Shakespeare was no friend of Catholicism, but that was a common attitude in
Elizabethan England. I don't think Tolkien had any problems with the
religious content of Shakespeare's plays as much as with his handling of pagan
legend and lore (Curious mentions Macbeth and Midsummer's Night Dream--two
plays that especially bothered Tolkien because of their treatment of witches,
prophesy, and elves).
We should also note that Tolkien was in part reflecting the bias of Oxford at
the time, the "Lang" vs. "Lit" battle that defined the study of English at that
institution. Language scholars who studied philology and literature up to
Chaucer (such as Tolkien) saw themselves in opposition to the Literature
scholars who look at post-1400s literature. In Tolkien's day as in the
previous century, many scholars were prone to excessive and sometimes overly
flowery Bardolatry, so Tolkien's cordial dislike of Shakespeare probably was
partly his curmudgeonly way of thumbing his nose at his Literature colleagues
who would say that there's nothing else worth reading *but* Shakespeare.
Tolkien was a crusty academic at heart, and they love to drop little bombshells
to irritate their peers whenever possible, so his attitudes were part of his
way of fighting turf wars at Oxford as much as a reflection of his own literary
tastes.
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"All wizards should have a hobbit or two in their care..."