Try adding this mime-type to your server. I added this to my .htaccess file:
AddType text/cache-manifest .appcache
It cleared up an error in the Mobile Safari Debug Console, but I'm 100% it enabled the caching in the Web App. I'm finding some accounts that say iOS Web Apps do not have access to the cache (which seems silly), but I'm still looking.
More details on the Application Cache and adding the mime-type:
Turning on the Debug Console:
According to this answer it may be critical to allow the app to fully download before saving to the home screen:
UPDATE: By 1) adding the mime-type to the server and 2) fixing any and all errors displayed in the Debug Console (which only show 3) after the entire site/app has loaded--including videos), I was able to get the cache working in the Web App.