The ?at= parameter normally returns the UUID of the name at the provided UNIX timestamp (in seconds, not milliseconds). If there is no timestamp provided the current time will be used. ?at=0 can be used to find the original user that had that username, but for whatever reason this only works if the account's name was changed at least once or if the account is unmigrated/legacy. This isn't even the main issue, although this should be fixed as well if possible...
Currently, the ?at= parameter seems to be broken, as it returns 200 (profile is found) for any timestamp provided, which shouldn't happen - for example, the account "MCB" seems to exist in 2010 according to the API (when I passed it the timestamp 1263146630) when in reality it was created in 2014. This issue should be fixed as there are numerous services that rely on this endpoint to provide user information, or help migrate old user records from servers.
This is my first public Jira report so I hope I've done a good job explaining this issue. If not, please let me know and I'll do my best to clarify.