I agree that 'The Alchemist' clearly is a juvenile story (i.e. the narrative technique isn't fleshed out yet and contains numeral 'mistakes') and also that at least the first paragraph consists of nice evocative sentences. I was curious what S.T. Joshi had to say about this tale, so I looked it up in his 'The Weird Tale'. He's actually quiet positive about this story because of how Lovecraft handles the supernatural in a (according to him) innovative way: first the 'typical' supernatural explanation seems to be a familiy curse (so, sorcery), but then the typical HPL-way of wanting to use a natural explanation causes the solution of a living human being, killing the aristocratic scions. Then, Joshi explains, Lovecraft puts the supernatural back in by having this alchemist to prolong his life unnaturally. I myself think that this latter was way to obvious in the tale (but yes, Lovecraft still had to learn better story telling techniques). On the other hand Joshi's take on it in itself (as a new way of approaching the supernatural) is interesting.
He also points at the typical Lovecraftian archetype of 'the very old man' which has its starting point in this tale (and indeed he has given one of his tales the name 'The Terrible Old Man'). I think this is something to go back to when discussing other HPL tales, like 'The Picture in the House' or 'Cool Air'.