I think Elementarys Sherlock is closer to the book version. In the BBC version he is totally aloof of social connections and norms, but in the books it is pretty clear, that Sherlock is able to tranverse London society - he had many case with high society people before Watson was part of his life - he just dislikes it.
If you're looking for a portrayal closer to the books, I'd highly recommend Jeremy Brett's Sherlock Holmes produced by Granada TV from the 80s.
I don't think any actor has come as close as Jeremy Brett did.
I tried watching an episode of the bbc version and found it very off-putting, it was written almost as a caricature of holmes. elementary was definitely a lot closer in spirit.
I've not read much of Holmes, so I can't really speak to the original character, but I would point out that the "he can socialize, he just doesn't like to" bit is somewhat part of BBC's Sherlock too - look at the relationship he developed with the woman to get at Magnussen. It's an aspect of him they never really explored much beyond that though - you're definitely right in that he seems more incapable of it than anything else.