Hey there–excellent question!
I honestly haven’t seen an example of The Events Calendar being used with an external API to deliver events. We are looking at ways to sync events with something like Gmail, but nothing quite yet. I believe that would be quite a custom undertaking.
I do wonder, though, how something like this could work using the WordPress REST API–considering that the external source you’re using would be WordPress as well, or if you would be feeding events from The Events Calendar to another source. Interesting possibilities for sure!
Sorry I don’t have more of a concrete answer here, but does this at least help? Please let me know. 🙂
Cheers!
Geoff