TL;DR - My template included third-party components which SharePoint refuses to have in its templates. I was forced to recreate the list.
When attempting to recreate a list using the template method, I found that the template did not populate. As a reminder, the normal process to create a list from a template is as follows:
At step 3, I found that my template was not listed. Where was it? I confirmed its presence in ~/_catalogs/lt/Forms/AllItems.aspx.
So, I downloaded the file to investigate its contents.
First thing, you'll find the extension is .stp. My first guess was that it was a glorified .zip file, but I was wrong. In fact, it was a glorified .cab file (like InfoPath forms). To open the file, I changed the extension to .cab and opened the internal .xml file. This file is really interesting and contains properties for the template. The main issue, however, was not in the properties but the fact that it was using a third-party component. Further testing showed that templates without those components worked fine.
I ended up recreating the list.