AMAZING NEWS: this problem is FIXED in iOS 8 for the Apple TV 3!
This problem has been driving me crazy for ages. I can confirm with pretty solid certainty that this is fixed on the upcoming iOS 8 GM for the Apple TV 3 (or equivalent, but that's what I'm calling it). I tested with the test file found on the page six of this thread (search for 'paisley' or scroll a bit) on the *current* public release of the Apple TV 3's software, although I had plenty of my own with the issue. Sure enough, I had the stutter. I hooked up the USB cable and restored to the GM of iOS 8 for it, set everything up and tried again. I've nearly finished the test file as I write this (24m out of 29m played so far without pausing or touching the remote except to check the time just now… it's an interesting show!).
This is for the 3,1 variant of the Apple TV 3. For those unaware, there's a silent hardware update with identical capabilities but a processor die shrink, which is the 3,2 version. I don't have one of these so can't test but as we all know this is a software issue since downgrading on capable units makes it go away, I imagine it'll be fixed on that too.
I'll report back if anything changes. Anyhow, for now: woooooooh! 😀
P.S. I spoke to the developer of a remuxing tool and the problem appears to be with remuxing with certain tools. FFmpeg has it, but mp4box doesn't — mp4tools has to use FFmpeg to encode audio, but the edit tab of that software uses mp4box, which fixes the issue on files with it. Still. If you're in the developer program, update away, and for the rest of you, hang on until Wednesday 17th (I think?) and it'll be released publicly.