Thank you for your responses.
Ajshort, after reading your suggestion, I thought it may have something to do with inheritance or the file system (a file missing, uploaded to the wrong location, or overridden by another file). To ensure this isn't a simple placement error, let me explain the structure I have.
I am viewing the HomePage, an extension of Page. In my themes/themename/ directory, I have code/HomePage.php, code/Page.php (where the code I posted can be found), templates/HomePage.ss, templates/Page.ss, templates/Layout/HomePage.ss, and /template/Layout/Page.ss. In essence, edits I make to the templates/Page.ss file are reflected both on the HomePage and all other pages, with the exception of the content ($Layout), which differs between the HomePage and all other pages. If this is confusing (or poorly written, as the case may be), I can perhaps upload a .zip file that more clearly explains where I'm coming from.
Assuming that the structure above is correct, I have been putting the Page_Controller class's extension of ContentController in code/Page.php. Ajshort, you pointed out that this works when rendering the controller -- if I am looking at the HomePage, am I correct in my assumption that the controller is, in fact, being rendered? If not, could please explain how I would create a class extension of the model instead?
Thankfully, this does not appear to be a caching issue. I would feel quite silly if it was! The Wiki poignantly noted that using a ?flush=1 query string would reset the cache. (I have found that shift+reload is necessary most of the time despite that, however.)
Sean, your last comment hinted at something somewhat unrelated, but interesting. I was under the impression that it was more pragmatic (and conventional) to use private and protected variables to prevent the illegal reading or overwriting of variables among classes and/or subclasses. Am I misunderstanding you, or is this structure unique to SS?