I had the exact same reaction when I came to this site a few years ago now. I also wondered what would be the best way to propagate
lichess.org to the masses.
It's compounded by the fact that whilst it may be fairly straightforward to contact people who play over-the-board chess, the majority of those people are elderly men who are skeptical of change. If lichess wanted to get the most out of advertising, it wants to target young, educated men in emerging markets, as lichess receives about 33% of it's traffic from Russia and Iran.
If you look at the Alexa figures, lichess is around the 17,000th most popular website on the internet globally, but it's the 3,700th most popular website in Russia. It's the 1,770th in Iran. I personally think compounding on this success is vital, and could lead to getting published in online news outlets in those regions (which then in itself helps lichess get noticed elsewhere).
Once I have more time (ie, after my summer law exams), I'm planning on going on a big PR drive. I've done PR work and have a good friend who did PR for an American start-up, so I'll ask her to help too in creating press releases, targeting online media (like HuffPo, Wired, Techcrunch, Forbes). I'll also probably politely ask as many GMs, IMs, and NMs as possible if they would like to check out the site. I'm sure there will be some who will agree with open source, free technology, and endorse the site out of kindness.
Ultimately, this goal is twofold - it creates a greater online presence just by being completed, but if it does attract any media attention, lichess could be included onto Wikipedia, which is then of course a great stepping stone for future PR drives.