Instead of Trigger Continue with operation after custom logic?
Posting this as an answer to address the core question:


  Is there a way to perform my history insert and then tell SQL Server to continue with the original INSERT, UPDATE or DELETE without writing those myself?


No, it is not possible to have an INSTEAD OF trigger just perform the original DML for you. This is why they are called INSTEAD OF and not BEFORE - they replace the original action with whatever logic you want to perform, which may or may not include DML against the original target table, depending on the scenario (you may want to prevent the insert instead of rolling it back, you may be migrating to a new table and using this as an interim solution, you may have distributed or otherwise not-directly-updateable views, etc). 

If you feel you have a valid use case for INSTEAD OF, then you will have to manually code the DML they're replacing. I don't think you do, so, as the others have suggested, use an AFTER trigger.