I've read this story a couple of times now and to be honest I don't like it much. But I'm going to try and find some positive things to say.
My main problem is that this story hasn't really aged very well. Maybe in 1905/1918 the idea of a man getting lost in a cave and becoming beast-like would have been shocking, but in 2019 it just feels a bit mundane or predictable. That said, I agree that Lovecraft has done a good job telling the story he has decided to tell - and for all my criticisms, it's far superior to anything I could write now, let alone when I was 14! And actually, having read a lot of Lovecraft before this, the fact that the "beast" was actually a man was surprising, if not shocking. I suppose I was expecting a Mi-Go or a Shoggoth (not that Lovecraft had worked out those sorts of things at this stage), although in the context of the story as it is only the revelation of it being a man would really work.
Additionally, the narrator is just way too ideal to be compelling: "look how amazing and intelligent and unemotional this guy is - he's not scared of anything!". I guess every author has been guilty of this at some point, and to be fair, Lovecraft at least had the decency to have his intellectually superior human give in to fear at the climax. But the story suffers from the early description of the protagonist. I like the comparison with the protagonist in "The Statement of Randolph Carter" being a corrective to this sort of thing.
I think the other Lovecraft story that this one called most to my mind was "Facts Concerning the Late Arthur Jermyn and His Family" (according to criticisms I've read, one of Lovecraft's most racist stories - surely an accolade of some sort, if not a positive one), which has the "white ape" trope played out the opposite way to this story (I won't say any more, so as not to spoil it for anyone who hasn't read it). For me, this comparison makes "Beast" seem a lot more nuanced - even if perhaps unintentionally, and only in the context of Lovecraft's later works. Rather than the revelation being of cosmic horror, strange alien creatures or impure bloodlines, or the people involved being "degenerates" (whether through breeding with non-humans, or just being races Lovecraft didn't like), the horrible, fear-inducing beast turns out just to be a man - a strange man, marked by his time in the cave, but a man nonetheless. The fear of the unknown that pervades Lovecraft's work is there, but is proven to be unfounded (we don't know whether the beast-man was intending to attack the narrator, but we're given no reason to think so). So actually, the true horror of the story is how a man reacts to his fear of the unknown with violence. Even today, this is an important commentary on human nature, but sadly it seems almost ironic when written by someone with Lovecraft's (deserved) reputation. And in the end I could have done with just a bit more sense of regret for his actions on the part of the narrator to fully convince me that Lovecraft was completely sincere in making this point.