I have had this conversation so many times over the last five years, and my response has not changed from then to now.
Yes, mobile sites matter. A LOT. After all, more than 50% of internet users accessed websites using a mobile device (excluding tablets). But wait, that leaves at least 40% of users who didn't. I am not going to list every statistic on mobile users vs. desktop vs. everything else because it varies slightly from report to report. Suffice to say, all devices still matter.
While the number of mobile users continues to grow year after year, there is still a huge base of users who prefer or utilize non-mobile devices (desktop users come in between 40-45%, according to current data).
Between Generation X and the Baby Boomers, both of which consist of millions of people who still prefer to browse, shop, buy, search, and socialize via a non-mobile device, it's as important to not alienate them as it is to appeal to the mobile-only generations.
This is why we don't focus on mobile sites first, or mostly on mobile sites, and why your designer should ensure that all devices matter (mobile, desktop, and everything in between), no matter how your site is built.