Is there a way to get out of this infernal loop?
Yes you can use this link to clear out your sign in information from our system. It should stop you from being redirected to the Welcome to MyDMV page.
Wouldn't it be easier if I could sign in using my personal NY government account?
As it so happens, New York State keeps your personal identity separate from your government employee identity. Therefore, you will need to create a personal account with a different username than the one you use for work.