You are wrong. ranking does not depend on number of backlinks but depends on quality of backlinks and those low quality backlinks would harm your sites seo.
21 high competition keywords means you have to be doing lot of content marketing and social media marketing.
Apart from that you should also be focusing on traffic from social media sites and come out of the spamming for backlinks concept as that was case in 2010 and not any more. If you get too many links from those sites, then you have to get those links removed if flagged in Google webmaster tools.