The “time machine” page is somewhat like what the newer API endpoints are like in that it doesn’t differentiate between available usernames and ones that are unavailable because of things like suspensions.
The v3 endpoint will give you exactly what you need. If the username is in use, you’ll get back the channel object, if it’s unavailable for some other reason, you’ll get back a 422 unavailable message, and if you get back a 404 not found then the username is available for use.