"We humans define intelligence in such limited terms." How would you define intelligence, then? Being able to correctly respond to human emotional cues has been evolutionally advantageous for dogs. They've had thousands of years of coexistence to refine their instinctual abilities to "interpret" our emotions, which leads to more attachment on our part, which leads to better reproductive opportunities for those dogs which we are more attached to. For all we know, dogs responses to human emotions, gestures, and nonverbal cues are a completely autonomic behavior.
I love dogs, I think my dog is super-smart, funny, and very sweet. I've heard her particular breed referred to as "the dog with the human brain" on some dog TV show before, and I'm sure she's not even as smart as a collie. That said, I can't help but feel like your comment is a bit... overwrought. Dogs clearly understand parts of human language and judging them by their ability to understand it seems like the best way to determine their intelligence. I would hardly call it "remarkably arrogant". You have to consider the fact that humans have likely had spoken language for as long as we've had domesticated canines.
In the end I can't help but feel like over-sentimental and hyperbolic (would you say we've "underestimated the intelligence" of, say, jellyfish?) declarations like yours really set us up only for disappointment. Our dogs are great, and probably do actually love us, but they're not furry little geniuses held back only by their lack of proper speaking ability.
All that said, they're still way better than cats.