My problem ended up being totally unrelated to the symptoms...
It appears to be some sort of bug with Crittercism running on iOS 7. While looking into a debug message in our console ("(webView:decidePolicyForNavigationAction:request:frame:decisionListener:)
failed to return after waiting 10 seconds"
) we came across this question: iOS 7 UIWebView not rendering, which suggests installing an updated version of Crittercism. After doing that, our problem with video playback seems to be fixed.