And yet it did. Not to mention all the other magical things the transporter has done over the years. Kirk being split into two (and no, he wasn't dying because each one only had half as many particles as the original), ages being reversed, characters being reset to older data, the existence of replicators which are based off transporter technology, and so on and so forth.
It's clear the transporter does destroy you and creates a duplicate at the other end; just because the duplicate also (usually) happens to have the same building blocks as the original doesn't mean it doesn't. Especially since there's literally -no- other way for it to work as depicted. None, zip, zilch, nada.
The only other way that I've seen mentioned in literature is for the transporter to open a temporary mini wormhole between the two points and push you through it.