Well, I was having this exact problem too, and like many here, spent a long time reading these messages. As I posted early in this thread, the problem is NOT unique to iOS 7- I was having it with iOS 6. It's not unique to the iPhone 5- I was having it with my iPhone 4S. Anytime battery charge was around 50% or less, any use of a high-draw item like the camera would cause an immediate phone shutdown, and then indicating it needed a charge.
I'm not an engineer, but there is no doubt in my mind that what we are seeing are lithium-ion batteries that are approaching the end of their useful lives. I was lucky and got about 20 months out of mine before I had symptoms, and yes I realize some users here are seeing less than that. Lots and lots of variables affect battery life- number of recharge cycles, depth of recharges and discharges, blah, blah, blah.
I was lucky- my phone was under AppleCare (which I always purchase) and the phone was replaced no-questions-asked.
Now, if I didn't have AppleCare and was currently experiencing this issue and wanted to actually get the problem fixed- I would simply take my phone to any local computer repair place that does iPhone batteries and have them replace the battery for about 50 bucks. Lots cheaper than a new phone and heck, cheaper than AppleCare was.
Ranting here might feel good, but it's not going to fix the problem. Would it help if Apple "admitted" that li-ion batteries wear out? Duh. What's the point? They're not going to pay for that...nor would Samsung or Motorola or anybody else. This is really only an issue because replacing an iPhone battery isn't the easy pop-out, pop-in deal that it might be on some other phones.
Anyhow, I'm just another iPhone user hoping to help some others who are having the same problem I was and actually want to get it fixed. Cheers!