I don't think that social signals are necessarily more relevant than other signals. For example, direct traffic, or people typing your site name directly into Google, are also influential.
I can't prove anything for certain, but here are a couple experiences I had:
A few years ago, I had a brand new site that got a mention in a front page article in a huge newspaper. Because newspapers can't link, the mention resulted in tens of thousands of people Googling my website's name within a couple days. That lead to a significant rankings boost that never let up.
I also had the same thing happen on social (different site) where one piece of content got a ton of shares all at once. Across the major social networks, we're talking 20k shares. Again, this was a new site, and again, search referral traffic shot up across the board within a couple weeks.
So I would say, yes, social can have an impact. But I believe that it isn't *necessarily* important if you can demonstrate that same level of interest in different ways.