I think that the use of computers by large businesses and other users was more common during the 1960s than the OPimagines.
As I remember, I actually had to design or write some mathematical calculation for a mainframe computer at my university about the time of Star Trek's last season. It took me several tries since each time I corrected an error I made a new error. And each try required a different visit to the computer center.
Scientists were already working on computer voices in the 1960s. I remember listening to a recording of a computer generated voice singing "daisy" back in the 1960s.
I also remember a science fiction story in Analog in probably the early 1970s when there was some sort of breakdown of all the computers in a small town. The characters didn't realize how many computers there were in town and all the things that thwy were used for until the computers started malfunctioing. This should have been before the first "microcoputers" (personal computers) so the computers would have been mainframes or minicomputers.
I remember wondering at the time whether the computer dependent society in the story was supposed to be a near future setting or the actual present.
A minicomputer, or colloquially mini, is a class of smaller general purpose computers that developed in the mid-1960s1 and sold at a much lower price than mainframe2 and mid-size computers from IBM and its direct competitors. In a 1970 survey, The New York Times suggested a consensus definition of a minicomputer as a machine costing less than US$25,000 (equivalent to $174,000 in 2021), with an input-output device such as a teleprinter and at least four thousand words of memory, that is capable of running programs in a higher level language, such as Fortran or BASIC.4
The class formed a distinct group with its own software architectures and operating systems. Minis were designed for control, instrumentation, human interaction, and communication switching as distinct from calculation and record keeping. Many were sold indirectly to original equipment manufacturers (OEMs) for final end use application. During the two decade lifetime of the minicomputer class (1965–1985), almost 100 companies formed and only a half dozen remained.
So mincomputers were begining to appear and be used in smaller businesses in the later 1960s.
Some older SF fans may remember the fake computer systems with countless blinking lights in Irwin Allen science fiction productions frome the 1960s. I read somewhere that they were actually real computers from the 1950s which were obsolete and were acquired cheaply by Allen.
Here is a link to a computer timeline indicating that the first business computers and first mass produced computers were manufactured during the 1950s.
And during the age of protests the message "do not fold, spindle, or mutilate" on computer punch cards was well known enough for the message "I am a human being do not fold, spindle, or mutilate" to appear on shirts and buttons.
In fact there was a TV movie with the title Do Not Fold, Spindle, or Mutilate as early as November 9, 1971, a mystery involving a computer datng service.
The first computer datng service was started in 1964 according to this timeline:
So the evidence I can gather at the moment indicates that while very few people personally used computers during the 1960s or were familiar with them, computers were starting to affect people's lives.