RedirectsI was wrong when I thought the redirect system would handle resource files like png/jpg/js/css/etc... Joomla is setup in such a way that the web server only passes on non-resource requests (like the pages and html) to Joomla. So, the redirect system never even sees the request for something like apple-touch-icon.png. It's not impossible to change that configuration, but then every missing resource would trigger the CMS, causing needless load.
Meta TagsIt looks like Safari was ignoring those <link> tags because they were inside of <div> tags (which would technically be invalid in the <head> portion of a website). This is something that mod_custom wraps the content in and there is no way to turn it off. To solve this issue and to help with other cases were custom tags are needed in the <head> portion of a website, I created a new module named Raw Tags. I copied over the touch icon tags with slight modifications into this new module on StVin-Omaha and can confirm it's now working on an iPad when using "Add to Home Screen".
<link rel="apple-touch-icon-precomposed" sizes="57x57" href="content/images/apple-icon-57x57.png" /> <link rel="apple-touch-icon-precomposed" sizes="72x72" href="content/images/apple-icon-72x72.png" /> <link rel="apple-touch-icon-precomposed" sizes="114x114" href="content/images/apple-icon-114x114.png" /> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="content/images/apple-icon-144x144.png" />