Why don't phones correct vertical filming?
There is indeed no technical limitation that would prevent us from preventing vertical videos in software. The issue is more of an UX/usability problem.
While the sensor would allow us to capture virtually any part of it (video has usually always a lower resolution than the sensor can provide, when making videos the phone just records a small portion of the sensor area) we kind of don't want an auto correcting orientation as its rather unintuitive. The emphasize is on kind of.

Your phone screen is 9:16 so in order to show 16:9 video on your phone screen you either need to turn your phone by 90° or show the video with giant black bars on the top and bottom, which would be very annoying when recording a video. It's quite unintuitive to record horizontally while in a vertical position, so from a UX design perspective you would say the user wants to shoot vertically when holding its phone vertically because thats what you would usually expect your phone to act like.
Hold it vertical you shoot vertical, hold it horizontal you shoot horizontal, a pretty intuitive concept.

Though its still no reason to not at least implement an option to auto correct this and my guess why phone manufacturers aren't doing this is probably largely due to increased engineering costs in the camera module (afaik it can't be done purely in software, you would probably need to alter the functionality of your camera module a bit).

Users who shoot vertically don't really care about the fact that they have a horrible aspect ratio as they mostly just watch their video on a phone anyway. Vine is a good example for that, it really promotes doing vertical video because the aspect ratio of the device that your video will be watched on most of the time has the same vertical aspect ratio.
So the demand isn't really that huge so far to warrant such a feature but I'm certain we will see this at some point.