The Biggest Mistake Most Bands Are Making With Facebook
Facebook should not be your sole tether. Because it is not even your tether to begin with.
With your own site or with an email list you are in control of what you post and how/when/who gets to see it. Whether your fans see your post about a new t-shirt design isn't contingent on whether they've "liked" you lately or on a site's algorithms; your communication with fans and the general population is not mediated by someone else's empire, nor dictated by the tides of social media. Having that control, that ability to maintain that connection as it best serves you and your band, is incredibly valuable. The smartest thing a band can do is utilize Twitter and Facebook and other social media to direct people to their own site. In the long run, all the "likes" in the world don't mean shit. Owning the means of production/distribution does.
That said, Facebook is an easy way to promote and invite people to your shows. It's an imperfect, kinda creepy, too-easy device that now seems absolutely crucial to bandhood--which is exactly how Facebook (and other sites) need us to think in order for them to survive. It can be incredibly useful, especially for touring bands and people interested in promoting their music who want that accessibility. I don't really want to say, "Yes, every band should have a Facebook page," but for most bands, it's probably a fine idea.
Whether and how your band uses social media really depends on what kind of career you are interested in having. Contrary to popular thinking, constant self-promotion doesn't have to be a fundamental part of music-making in 2013. To paraphrase good ol' Ian MacKaye, if people really want an alternative, they will dig (or Google) until they find it. It is fine -- liberating, even! -- to live and make art, as you've termed it "sans Zuckerberg," and get by on word of mouth and flyers. I mean, that's how bands have promoted shows ever since Shadrach, Meshach and Abednego started playing out -- it still works.
To address your other issues: Is it possible to get heard without putting your music online? Yes, but this is the primary way that people -- especially young people and people interested in new music -- consume it. There are surely bands that do not use the Internet, but it's hard to imagine "not using the internet" would be cool enough to gain foothold as a wide-scale trend. And, yes, you are old; getting bent up over the last little bands on Dischord having Facebook pages seem like a poor use of anyone's brainpower. Also, "pimp" is a pretty loaded word in relation to a band fan page. The world is a different place since Six Finger Satellite's heyday, thank fucking gosh for that. Being discerning about a band's big picture morals (or lack thereof) and how they navigate their relationship to their fanbase is even more crucial now that corporate tie-ins are the pervasive norm -- but using social media is a relatively minor offense on the selling-your-soul-for-a-chance-at-the-bigtime scale.