It would be a dopey workaround but you could have a task at the end of the sequence that's a dummy task, and just don't complete it. Or create a new activity status that means completed to you but isn't the same as the core completed.
If you're up for writing a small extension you can implement https://docs.civicrm.org/dev/en/latest/hooks/...
If you want the tokens only in the context of a schedule reminder, you should use the new better way to do tokens instead : TokenProcessor.
hook_civicrm_tokenValues is only working for Contact entity. You can't really add tokens on other entities like CiviCase using this hook.
If you need the token to work in every contexts (transactional email, mass ...
Pradeep's answer should work. But there is actually an entry in civicrm_log where the modified_id is the true logged in user even if you change the Reported By field.
But I don't know if it's usable in Views.
You could also create a custom field for activities to store the logged-in user.