Does the use of English words weaken diceware passphrases
I don't believe that looking at the diceware phrase is very helpful.

Instead, look at it this way; there are $7776^7 \approx 2^{90.47}$ possible 7 word diceware phrases, and diceware will select one of those possibilities uniformly.

That means that the probability that one specific phrase is selected is at most $2^{-90.47}$.  Or, in other words, if someone were to guess random diceware phrases, he would make an expected $2^{90.47}/2 = 2^{89.47}$ guesses on average before hitting the right one.


  Is it not the case, however, that using English words in place of the underlying dice-throw numbers reduces the entropy of the passphrase.


No; there is a one-to-one mapping between diceware throw numbers and valid diceware phrases; one can take the phrase, and map it back to the throws that generated it.  And, a one-to-one mapping never effects the entropy.

What it does effect is the 'entropy-per-character' measurement; because the diceware phrase is (usually) lengthy, the 'entropy-per-character' measurement is a bit low.  However, we don't really care about the 'entropy-per-character', instead, we care about the security that the entire passphrase gives us.

Remember, the 'entropy of a passphrase' is a bit misleading; if we talk about entropy, we need to talk about the process that selected the passphrase.  In diceware, we have a very different process than if someone were writing some English text (and so the entropy estimates that may be valid for human generated English text do not apply to diceware generated texts)