There's no good reason why cameras can't do a better job at video.
But first, a qualification: I'm not talking about shooting full-res 20MB 16 megapixel frames at 60 FPS, which the other answers point out is hard / impossible for now.
But I see no reason why cameras shouldn't be able to do a BETTER job than they do now, which is shoot 60 FPS at anything better than 1080p, like 1440p (2560 x 1440). Or video at a frame rate greater than 60 fps, and any decent resolution. Heck, the iPhone shoots 120 FPS at 720p.
And if we compress that down to 30mbps (< 4 MB /s), SD cards should be able to handle that comfortably. Phones are starting to support 4k video, and Qualcomm's new mobile SoC handles 4k video encoding, so it's not a hard thing to do. And standalone cameras usually have bigger batteries than ultra-thin phones, so power is not the constraint, either.
I was told that iPhones have more powerful processors than standalone cameras, but I see no reason why that can't be improved. The SoC in the iPhone 5S is estimated to cost only $19. When people pay more than a thousand dollars for an NEX with lenses, $19 is a trivial amount more to pay if it results in better capabilities.
Like you, I own an NEX, which cost me a substantial amount of money. Don't get me wrong, I like my NEX, and I'm not saying, of course, that the iPhone is better overall than the NEX for photography; just that it's better at SOME things. And I see no intrinsic reason why the NEX can't do everything or most of what the iPhone 5s can do:
Take 720p video at 120 FPS.
Take video at a higher resolution than 1080p.
Take a burst of photos and merge them in software to reduce noise. This is called image stacking.
Have a panorama mode that works. The NEX's panorama mode, at least on my NEX-5R, is useless. It says I'm moving my camera too fast, or too slow, or not in a perfect line, or something else.
Tap to set exposure for a certain part of the scene. Maybe the NEX-5R supports this, but the touch-screen is so bad it's not worth using.
Again, my point is not that the NEX sucks (I wouldn't have bought it if that were the case) but that there are SOME things the iPhone does better, and I don't see these as intrinsic tradeoffs.
If you're talking about putting an APS-C sensor in a pocketable camera, that's probably an inherent tradeoff for now. If you're talking about making a 50x zoom lens for an APS-C sensor camera at a reasonable price, size and weight, that's an inherent tradeoff, again. But I don't why the things above are inherent tradeoffs. They can be fixed with software, and a more powerful processor. We're not talking about any fundamental limitations of optical engineering.
So, my answer to your question is not something you'd perhaps like to hear: there's no reason (standalone) cameras can do a better job at some of the things you're asking about.
1 Yes, it takes more power to process the output of a 16MP sensor (like the one used in the NEX) than it does to process the output of a 8MP sensor (like the one used in the iPhone), but we can easily get around that by reading only some lines from the sensor.
2 Yes, NEX users expect better quality in their videos, but the least I'd expect is that my NEX at least does what my iPhone can: 120 fps at 720p, with comparable quality. If it can do better, great, but it should at least be as good as my iPhone. Saying "we can't do a better job than the iPhone, so let's not even try doing as good a job as the iPhone" is not excusable.
(Credit to Michael for raising these two questions.)