Personally, I think 'believing in the product' is overrated. If you emphasize that too much, you end up with a bunch of yes men who can't see (or at least won't mention) problems with your product. Overconfidence is dangerous. The ability to be critical of yourself and your company is essential to avoiding death, I think.
That said, there is no way I'd hire a customer service person who wasn't a customer first. (but then, I sell a product that is of interest to technical people, and I target the low end, so really, #prgmr on irc.freenode.net is probably a fairly decent place to pick up cheap SysAdmins.) If I sold to, say, real-estate agents, I might feel differently. Also, if I was looking for an MBA, my customer pool would probably not be a very useful place to fish.
Really, people I know, and people who know people I know have been a more useful resource than anything else. (I don't mean people on my "social network" in the sense of facebook, but, for example, I met the co-author of my book because he was the roommate of one of my employees.)