Create a view that shows the replicated events in a tab

They have done some interesting things here
https://www.drupal.org/project/recurring_events

I prefer your approach of centering it on date_recur.
That way any entity can use the entity_repeat module.

If at all possible it would be nice to be able to add an entityreference to an entity that is entity_repeat enabled.

Possibly create an api hook or add to the current ones so that the original entity is passed in.
you can then add/append the newly cloned entity to the entityreference field.

Or create two entityrefence, entity_repeat_original_ref, for the original entity
entity_repeat_cloned_ref, for the cloned ones. Keep adding the original ref to each of the clones. We would just have to make sure that this action populates entity_repeat_orignal_ref.

This second way is flexible and can handle the use case of what to do if you use one of the new clones as the base for a different entity_repeat.

But should you be able to delete the original entity that has entity_repeat_original_ref?

I think entityreference already takes care of recursion but would have to doublecheck