Hi Jon,
Actually an event template record is also an event record, event template record is just controlled by field "Is template".
Do you mean you can't see any event template records in event template selector dialog(2) even if you had created an event template record(1)?

Please check whether the record is actually saved in system.
1. As we see, even default All events view will filter out template type records.

2. Create a new custom view which hasn't any filter criteria, and add Is template field into the view, save it and publish all customization.

3. Then under the view, we can see which event record is template type.

4. If template record exists but you can't see it in dialog, it might be you still don't have enought privileges to export template,(even if it's same type entity)
please assign necessary privileges(full level is recommended) to those entities for your current user role.
* Marketing data import
* Custom Control
* Grid
All of them are in Custom Entities tab.
Regards,
Clofly