I seem to have resolved the battery drain issue on both my iPhone 5s and on my iPad 4, but I am not sure that my story will be of much comfort to those of you struggling with the problem.
After installing 7.0.4 on the two devices, I found that the phone lost its charge quickly (15-20% an hour doing next to nothing) and was warm to the touch. I then noticed that both iPhone and iPad were showing that Usage and Standby were virtually identical (look in Settings>General>Usage>"Time Since Last Full Charge" - if you haven't been using your phone constantly, "Usage" should be a time considerably smaller than standby). That meant that something was keeping the devices from ever going into standby.
Over four days, I had two online chats and one phone call with Apple and made two Genius Bar visits. We did every sort of restoration to factory condition with restore, clean wipes with setup as new phone, etc. etc. Nothing helped. Ultimately the gave me a new phone which ran 7.0.3 (and suggested that I not upgrade to 7.0.4), and did a DFU restore on the iPad (which they did not really seem to think would help - last ditch effort).
The phone was set up as a new phone running 7.0.3, but it *still had the battery drain/usage=standby* problem! After much experimenting, I figured it couldn't get much worse, so I upgraded it to 7.0.4. Now I get a decent standby out of it. Problem solved.
The DFU-restored iPad didn't get any better, so I ultimately restored my backup out of convenience - as long as I have battery drain, I might as well be able to use it normally, right? For a day and a half it continued to have the no-standby/battery drain problem, and then today it started working properly again! We'll see.
The online advice that I can give is that if you've got a phone with this problem, you should be patient with the nice Apple people and let them try to fix it; if they can't, ask for a new phone. These things are too expensive for you not to be able to use them in a normal fashion. It's a frustrating process, but I've been reading all of the newsgroups for the last week, and no one has any good suggestions for fixing it yourself. Put your effort into getting the Apple people to fix it or replace it quickly so that you can go out knowing that you won't have to worry about charging your phone constantly.
If you've got an iPad that has the usage-same-as-standby problem, it's probably draining the battery quickly, too, but if you can survive, there could be an iOS update soon that would presumably address the issue. Of course, it's probably bad for the device to be churning away constantly.
Very irritating, huh? I spent a week dealing with that! I must say, though, that all of the Apple people I dealt with were exceedingly gracious and that one of them was actually knowledgeable and not just following a series of procedures. Only one of them pulled that "Turn down the brightness on your screen and turn off location services" nonsense on me, and when I called her on it, she made sure that I'd get to talk to someone who might actually be able to help me.