I think everyone here should agree that crosstraining is critical. Let's face it, all of the professional fighters crosstrain. They have to.
Every martial artist understands that styles have inheret weaknesses. A pure grappler could be in a lot of trouble against a really good striker. A pure striker could be in a would of trouble against a good grappler. Even the purest, most traditional martial artists have to concede that you must train for all situations, or you are not complete.
What if a striker got tackled from behind or the side and was on the ground before he got a chance to strike? What if a grappler (and his friends) are outnumbered? It would be silly for two or three grapplers to try and take down three or four guys, so they must know some striking to protect themselves. It would be difficult for a striker to escape even a freshman collegiate wrestler on the ground, so they must know some groundwork to at least get back to their feet.
Do grapplers and MMA guys think they are better than everyone else? I don't believe so, I believe that they feel they are more complete for real situations that more traditional styles. And they believe they prove it in the UFC, Pride, KOTC, etc.. To be honest, I think they do prove it. People say that the UFC, Pride, etc. are not street fights and that is true, but they're probably as close as you can come without pitfighting. I don't think anyone really believes that if you saw Vitor Belfort walking down the street that he would be vulnerable because he's "out of his element". Think that's crazy? That's the argument I hear though. "They need the cage to win", "They have rules in the ring so it isn't the same...". Whatever. They train to fight, and their arena is the televised competition.
Bottom line : Crosstrain. Crosstrain. Crosstrain. Don't sell yourself short by thinking you're never going to get taken down. Don't sell your students short by telling them if they train hard in their art no one will take them down and out of their element. Don't sell yourself short by thinking you're never going to have to throw hands, or you're never going to run into a guy you can't take down. Grapplers don't like to get punched in the face, so they train their stand-up game. Strikers don't like to be choked and armlocked, so they should train their ground game. Anyone who doesn't believe in crosstraining will fall victim to someone who does.