See dmr's paper at <https://www.bell-labs.com/usr/dmr/www/hist.html> for details, but in short exec and its equivalents elsewhere have always overlaid the running program with another program.  Early versions of PDP-7 Linux used the same process model as Tenex: one process per terminal which alternated between running the shell and a user program.  So exec() loaded the user program on top of the shell.  Indeed, this wasn't even a syscall; the shell itself wrote a tiny program loader into the top of memory that read the new program, which was open for reading, and jumped to it. Likewise, exit() was a specialized exec() that reloaded the shell.  The Tenex and Multics shells had more memory to play with and didn't have to use these self-overlaying tricks[*]: they loaded your program into available memory and called it as a subroutine, which accounts for the name "shell".

So it was the introduction of fork(), which came from the Berkeley Genie OS, that made the current process control regime possible.  In those days, fork() wrote the current process out to the swapping disk and set up the process table with a new entry.  For efficiency, the in-memory version became the child and the swapped-out version became the parent.  Instantly the shell was able to run background processes by just not waiting for them, and pipelines (once the syntax was invented) could be handled with N - 1 processes in an N-stage pipeline.  Huge new powers landed on the user's head.

Nowadays it's a question whether fork() makes sense any more.   "A fork() in the road" [Baumann et al. 2019] <https://www.microsoft.com/en-us/research/uploads/prod/2019/04/fork-hotos19.pdf> is an interesting argument against fork():

* It doesn't compose.

* It is insecure by default.

* It is slow (there are about 25 properties a process has in addition to its memory and hardware state, and each of these needs to be copied or not) even using COW (which is itself a Good Thing and can and should be provided separately)

* It is incompatible with a single-address-space design. 

In short, spawn() beats fork() like a drum, and fork() should be deprecated. To be sure, the paper comes out of Microsoft Research, but I find it pretty compelling anyway.

[*] My very favorite self-overlaying program was the PDP-8 bootstrap for the DF32 disk drive.  You toggled in two instructions at locations 30 and 31 meaning "load disk registers and go" and "jump to self" respectively, hit the Clear key on the front panel, which cleared all registers, and started up at 30. 

The first instruction told the disk to start reading sector 0 of the disk into location 0 in memory (because all the registers were 0, including the disk instruction register where 0 = READ) and the second instruction kept the CPU busy waiting.  As the sector loaded,  the two instructions were overwritten by "skip if disk ready" and "jump to previous address", which would wait until the whole sector was loaded.  Then the OS could be loaded using the primitive disk driver in block 0.