As to the "brushed aside" destruction, it's mostly because it just feels like it's done for the cool factor and that it's just all very casual, much like urban destruction in your typical Hollywood fare (Avengers, Transformers, Batman, etc). It's not like it's anything new to fiction where the main characters are invested much more in their personal melodramas when there are thousands of lives in jeopardy, so you're right in that. Still, it just feels odd.
We never really got the full aftermath of the loss of six billion Vulcans, for that matter. In some ways, that bothers me more.
I will say this about movies that destroy cities or large parts of them (and apparently we'll see some more of that in "The Man of Steel"), post-9/11 I don't think there's a "cool factor" to it, any more. It does feel odd, too. It's also too sanitized, given what we saw in New York. Where are all the clouds of dust? The fires? The rush of air and the flow of debris and smoke running through the streets for blocks and blocks? In STID, the Vengeance finally stops, and Khan jumps off into clear blue SF skies and exits down clean streets. We know it wouldn't be that way. It's so much messier. Much more horrible.
If they didn't want to linger on it, as Spock left for the Enterprise, they could've at least done a shot back to the path of destruction. Just a wide shot over Spock's shoulder as he glimpses back at it, himself. Something to show this was not just comic-bookish plowing through buildings for the "cool effect," but there were real consequences. It would've made the memorial scene later even more poignant.
Just my opinion.