Some "commonsense truths" have only tentative hold on reality and can be traced to a particular popularizer. The myth that "spinach making Popeye have super-strength is due to a misplaced decimal point in iron content data", for example, can be traced to one particular lecture (Professor Arnold Bender at the University of London in 1972). However, with the blinking 12:00, this really was a widespread circumstance, so we can't point to a particular person who made a catchphrase. We can instead accuse a particular piece of technology of starting the interface in question.
Consumer VCRs went back early, but didn't originally have the blinking display. This ultra-expensive one from 1964 was for consumers but only moved at most a few hundred units, and the Philips N1500 from 1972 had a clock but was an analog model.
By the late 70s you could have the legendary "12:00" display, like in this Panasonic model from 1978.
The HR-756OU from 1982 includes quite elaborate timed recording. It had the ability to record 8 times in a two-week period, so that the user could go on "vacation" (as the manual puts it) and find when they returned all their shows were recorded to "view at your leisure". I think this description of the process of recording a show (from the magazine High Fidelity, August 1982) gives a hint as to why the process caused so many issues:
Pressing SET consecutively advances the electronic tuner through the channels until you get to the one you wish to record for Program 1. Select switches you to the days of the week, Sunday first. Consecutively pressing SET advances the display through the days of the first week and into the next (indicated by the legend "2nd"). When you've chosen the day, you press SELECT, which switches the timer to the hour mode, and SET, which advances the hour. The next press of SELECT switches the display to minutes; with SET you advance to the desired starting time.
Setting the time in general goes through the same process, and for a long enough power failure, "the clock resets to 12:00 am Sunday and blinks to advise you that the programming has been erased."
This type of interface -- all the way up to having to keep pushing buttons to swap between hours and minutes -- became common enough that Newsweek, writing in 1992, was able to write about how
We can't even control the technology in our own homes, which is why our VCRs are still blinking 12:00 ... 12:00 ... 12:00 ...
Essentially, the same process of setting 1982 models held all the way to the 1990s, enough so that putting tape over the display was an extremely common "solution" to the issue. Guy Kawasaki (writing in 1990) mentions people who made money whenever Daylight Savings shifted by being paid to change times on VCRs. There were estimates in the late 80s that as many as 80% of owners could not program their VCRs.
How did this situation happen? This seems to be a facet of the "technological inertia" problem often referenced in business tech history. Essentially, business are loathe to change things if they're "working". The design of VCR clocks, which was fine with tech-savvy early adopters in the late 1970s when the displays came out, was held onto quite stubbornly.
This was a famous enough issue that it later got dubbed by Neal Stephenson (of Snow Crash fame) as the "blinking twelve" problem and it gets applied to engineering design issues in general, of settings too complicated for a user to want to change so they get ignored. This is still a serious issue in UI design, even if not for blinking VCRs in general; as Paul Schwartz writes,
Many or even most people will never comprehend the nature of the different graphical computer interfaces that are all that stand between them and the surrender of their personal data.
...
Greenberg, J. M. (2010). From Betamax to Blockbuster: Video Stores and the Invention of Movies on Video. MIT Press.
Schwartz, P. M. (2000). Beyond Lessig's Code for Internet Privacy: Cyberspace Filters, Privacy Control, and Fair Information Practices. Wis. L. Rev., 743.
Secunda, E. (1990). VCRs and viewer control over programming: An historical perspective. Social and cultural aspects of VCR use, 9-24.
Sutton, M. (2016). How the spinach, Popeye and iron decimal point error myth was finally bust. HealthWatch Newsletter, (101), 7.
"I can't get my VCR to stop blinking 12:00" was a staple Johnny Carson Tonight Show joke in the 1980s. Jay Leno (a younger who replaced Carson in 1992) remarked while a guest on the show that ever since he gave his parents a VCR, his phone bill had gone up because he kept having to explain how to set the VCR clock. And there was a widespread joke that the easiest way to get your VCR to stop annoying you by blinking "12:00" was to cut out an appropriately-sized piece of card and tape it over the VCR clock display. But in 1990, President George H.W. Bush quipped to a television industry audience that in addition to the six policy goals set forth in his recent State of the Union address, "I want to add a seventh goal: by the year 2000, all Americans must be able to set the clocks on the VCRs."
So observations about inability to set VCRs became widespread because, as you observed, it was used a lot in media, and it also resonated with an actual experience people had. Not being able to set the VCR clock has been described as a socially-acceptable incompetence: it's so hard to do this that you don't mind letting people know it and may even brag about it. Because setting a VCR clock was pretty tough!
Let's talk about the mechanics of setting a VCR. Setting a VCR clock typically involved some arcane combination of button presses on the VCR remote control or in some cases the VCR unit itself to enter clock set mode, set the time (and in some cases day and date) then return to normal operation. VCRs typically did not have any battery backup for the clock so if the VCR was ever unplugged or there was even the briefest power failure, you'd have to set it again. A VCR whose clock was not set blinked "12:00" to alert the user. So it was indeed somewhat difficult, certainly more difficult that operating other household electronics and appliances.
Why do all this? Many users didn't need to. Keep in mind that setting the VCR clock was not actually necessary to play a tape or to record something that you were watching, and surely most households had many other timepieces. Strictly speaking, having a correctly-set VCR clock was only needed for "time-shifting," which is what in-the-know people called setting the VCR to record a program at the time it was broadcast so that you could watch it at a more convenient time. (The legality of doing this for personal home use was upheld by the U.S. Supreme Court in 1984.) Practically all home VCRs could be programmed to tune in a particular channel and record at a specified time, thus technically enabling time-shifting. On the other hand, it was hard to do. Programming the VCR required not only that the clock be set but that the user do more complicated button-presses to schedule the recording.
In fact, many users didn't bother and apparently didn't lose functionality that they desired. Market research showed even though VCRs were increasingly common in American households (about 75 percent of households with televisions in 1992), time-shifting did not become more popular--in fact, as VCR penetration increased, time-shifting per household decreased. Early VCR adopters were perhaps more technologically savvy or had purchased VCRs with time-shifting in mind. Later VCR adopters, like Jay Leno's parents, may have received VCRs as gifts or only wished to use them for simple functions like playing tapes or recording the programs they were watching.
On the other hand, it's also possible that more users would have time-shifted if the technology had been easier to use. In 1999, Neal Stephenson reported the phrase "blinking twelve problem" to describe the loss of intended functionality because of problems with the interface. Although engineers had created these marvelous, affordable home entertainment devices that would let users view their favorite programs whenever they wished, they were too hard to use and so this potential went untapped.
But in any case, it seems to be the case that many if not most Americans households ca. 1990 didn't bother keeping their VCR clocks set.