3rd party plugins don’t work with GitHub Pages so that’s likely the issue. They only allow a select few due to security reasons.
You can use the plugin but it’ll have to be if you self host elsewhere or use something like Travis CI to build your Jekyll site and push the contents of _site back to GitHub to host.
You can still use last_modified_at, you’ll just need to manually update that timestamp in your post/page’s YAML Front Matter. Depending on the text editor you use there might be options. For example I use Sublime Text and Visual Studio Code, and both have extensions/macros for quickly updating a timestamp.