Why would you want to make manual work for someone who just wants their account deleted? You're possibly better off offering an option in the delete flow for them to "talk with you to see if you can work something out" versus manually processing deletion requests.
Effort on those requests might recover some users which may be especially valuable if you are a subscription business. If you can't benefit from interaction then immediately imitating deletion from an API seems the only thing that would pass muster.
I think different use cases will call for different solutions. My use case is a relatively tiny number of users and any manual work they would generate for account deletion would be nil, or very close to it.
It's not necessarily about recovering users who want to leave but rather minimizing the effort required to implement a more complex deletion flow that has a high probability of never being used by real users (in my case).
Effort on those requests might recover some users which may be especially valuable if you are a subscription business. If you can't benefit from interaction then immediately imitating deletion from an API seems the only thing that would pass muster.