Authorship and microformats (schema.org specially) would fall inside "making sure information is crawable" - that is, purely technical aspects.
The relation between sharing in social media and higher ranking on SERP is weak to nil in my experience (2m+ unique visitors site, thousands of Facebook likes in each post, low organic traffic).
I can't speak to your specific case, but from my understanding:
Bringing in traffic from social media translates to search rank indirectly: eg. if your site hosts user generated content. So people visit your site from social media, create and update your content. That content gets crawled and ranked.
If your visitors are not creating content, or if maybe that content is not being indexed properly, then it would make more sense as to why your site is not getting tons of organic traffic.
In some cases, your app/site is not designed for people in a searching mindset eg. "Xs near me" "cheap/new Y" or "who/what/when/where/why/how is/did Z". Therefore search is less important to you.
The relation between sharing in social media and higher ranking on SERP is weak to nil in my experience (2m+ unique visitors site, thousands of Facebook likes in each post, low organic traffic).