Going from “AT&T made phone switches” to the idea that Unix was intended to drive phone switches is quite a leap. The creators of Unix described its creation and development in some detail, e.g. in The Evolution of the Unix Time-sharing System:
What we wanted to preserve was not just a good environment in which to do programming, but a system around which a fellowship could form. We knew from experience that the essence of communal computing, as supplied by remote-access, time-shared machines, is not just to type programs into a terminal instead of a keypunch, but to encourage close communication.
So Unix was really a scratch-your-own-itch development at first, and intended to be multi-process and multi-user from the beginning. Of course the costs involved in developing it (mostly buying hardware) had to be justified, which quickly led to the development of text-processing software and various other developments later on.
You can run V1 Unix on an emulated PDP-11 on a modern computer; the default configuration supports 8 logins. A partial reconstruction of the original PDP-7 Unix is also available. The latter seemingly supported multiple users and processes, but only one of each at a time (ten process slots were available but only one process was swapped in at a time); that analysis is based on the source code as preserved in the reconstruction, but other recollections (closer to the time) differ. Thus Steve Bourne wrote in The Unix System V Environment
A cast-off PDP 7 with a 340 display was available but the PDP 7 provided only an assembler and a link editor. One user at a time could use the computer, each user having exclusive use of the machine. This environment was crude and parts of a single user UNIX system were soon forthcoming. The space travel program was rewritten for the PDP 7 and an assembler and rudimentary operating system kernel were written and cross assembled for the PDP 7 on the GECOS system. This early system did not provide time-sharing. indeed, much like the modern personal computers, the PDP 7 hardware was simple and provided no support for such activities. An assembler and a command interpreter were soon available. This file system provided a name structure that was a directed graph. A single directory was used for all subdirectories and links made through this directory.
Cross assembling meant using two computer systems and carrying paper tapes of programs from one to the other each time a change was made. The system was soon bootstrapped onto the PDP 7. The process creation primitive, fork, and process images were added to the system during this rewrite. Essential utilities, such as file copy, edit, remove, and print were soon available. This system supported two people working at the same time and the term UNIX was coined by Brian Kernighan in 1970.
Thus it appears that the system on the PDP-7 was initially single-user, but that was a short phase during the initial development, and eventually Unix on the PDP-7 supported two users simultaneously; this also matches Ritchie’s recollections in the paper linked above:
Processes (independently executing entities) existed very early in PDP-7 Unix. There were in fact precisely two of them, one for each of the two terminals attached to the machine.
Presumably the system was still limited to running a single process in memory at any given time, even if two users could log in in parallel. Before Unix moved to the PDP-11, its developers had even managed to build two B interpreters (including one with virtual memory) and used that to write a few utilities!
Diomidis Spinellis’ Unix history repo contains source code for most historical Unix releases, including PDP-7 Unix.