Sure, if by “you” you mean universities, Sun, DEC, IBM, etc.
As a non corporation end user with a sublicense from one of the above licensees (which is the apt analogy for an individual user in my opinion) you most certainly did not receive the source or have the rights to it.
I regard "traditional Unix" as the 1970s era. There were no sublicensees. You bought the source code from AT&T. That was the only option.
You then likely hacked the source code to make it work on your machine.
I can understand that others might include the commercialization of the 1980s and the Unix wars as part of "traditional Unix", but even then, my IRIX system ca. 1990 supported end-user kernel extensions, to handle new types of devices. Our imaging hardware was not "connected with well-defined text-based interfaces".
As a non corporation end user with a sublicense from one of the above licensees (which is the apt analogy for an individual user in my opinion) you most certainly did not receive the source or have the rights to it.