I think TC is actually quite open to new startups emailing them. The key is to be extremely concise and matter of fact. If you send them a long email full of marketing talk, their eyes will just glaze over and they'll give up after third sentence. (This holds for investors too.) But if you're (a) doing something genuinely good, and (b) you can explain it clearly, you're doing them a favor by telling them about yourself.

So tell them in the plainest possible terms what people can do, thanks to your startup, that they couldn't do before, and try to keep it to three paragraphs or less. I do this myself when I introduce startups to anyone, regardless of the strength of our relationship with them.