[PatrickHenry]

The trick is that "mysticism is important" and "religion is important" are logically equivalent once you define religion as equivalent to mysticism - If "religion=mysticism", then one proposition is what you might call a suitable paraphrase of the other.

Close. But he never comes out and directly says that mysticism equals religion. If he did, you'd be right, but the example would be too trivial to concern us. So maybe it's a quibble, but I don't think this example involves begging the question. At least it doesn't seem to be a particularly clear example of this.

On the other hand, my analysis isn't entirely justified by the text either; because McTaggert never specifically says that mysticism is a component of religion (which it almost certainly is, at least in the West). Instead, he ambiguously says "religion never remains for long altogether untouched by mysticism." Whatever that means.

This example may ultimately involve the fallacy of amphiboly, because of the lack of clarity in the argument. It really is terrible writing.

[general_re]

I think my analysis works just as well if we change it to read "religion implies mysticism" rather than "religion is mysticism", and that's rather easier to derive from the text, although I agree it's a mess in terms of clarity of meaning. But no matter what, "religion never remains for long altogether untouched by mysticism" means that religion and mysticism are always connected somehow - where there is religion, there will always be mysticism, is the only way to read "religion never remains...untouched by mysticism". Whether that's because mysticism is seen as a component of religion, as your analysis suggests, or because it's a consequence of religion, as mine suggests, really matters not - either way, it's a fallacy. That's why I liked your approach, because it covers the other of the two possible interpretations that I see. And with some minor tweaking, your approach will cover the final route of escape - rather than suggesting that mysticism is a component of religion as you have done, suggesting that religion is a component of mysticism will also lead one into fallacious reasoning, the explication of which is left as an exercise for the reader... ;)