I am going to Session | New Connection | instance | Database | Session Browser | Start Trace | End Trace | Confirm | Yes | TKProf Wizard | Next | Next | Finish.
This opens the tkprof'd .out file in the defined editor (i.e. Notepad). What I wanted was the raw .trc file. I do not have access to the database server's file system, so I cannot manually retrieve the .trc files from the udump_dest directory.
Yes, the .out file generated reports the "Trace file: C:\Users\<user>\AppData\Roaming\Quest Software\Toad for Oracle\User files\<instance>_ora_<pid>.trc".
Toad can extract the .trc file from the server and executes the tkprof locally (default option).