Why is a just a rather odd wh-word. Its distribution is very limited -- it can only have the word reason as its antecedent, and since it's never the subject it's always deletable. Consequently it behaves strangely, as you and others point out.
............
I think you're asking the wrong question, Zebrafish. Why is a wh-word, and they have their own category. They sometimes act like conjunctions do, but so do many other words.
Asking about the Part of Speech of some grammatical word is a waste of time, because the Parts of Speech were designed for describing inflected languages, and English is uninflected, which means there are a lot of little words lying around that don't have meanings or general purposes, but rather fit specifically in some construction to make it work. Wh-words are one such group of words.
............
The difference between an adverb and a subordinating conjunction is not very substantial. Most subordinating conjunctions introduce adverbial clauses and many if not most can also be used as adverbs with the same sense, without a clause (but often presupposing one). That's the issue here -- POS are important in a Big Bag of Words kind of grammar, where all you see is words and tags on them. But that's not how English works -- English is all phrases and clauses acting as adverbs or nouns or adjectives or verbs -- so POS gets confused between the clause and its head word. J Lawler: comments preserved