Hi, Matt.
That looks like a highly-customized version of our plugin. It doesn’t even have much of the HTML markup that our List View typically does.
Our List View typically looks more like http://wpshindig.com/events/list/
Regardless, you could customize the default List View by referencing our Themer’s Guide and, for example, creating a button for each event to each event’s iCal link, which is visible at the bottom of the single event view, like this:
These single event calendar export links are generated by the single_event_links() method in /wp-content/plugins/the-events-calendar/src/Tribe/iCal.php
If you need some coding help, you may want to ask your developer or reference our documentation and list of known customizers.
Please let me know how this goes for you.