While it is hard to disprove the claim that no one made the connection between AD 666, the Mark of the Beast, and the end of days, we can safely say that their fears were not widely shared and that they would have been well outside the mainstream of those who concerned themselves with the calculation of such things.
One important reason for this is timing. Dionysius Exiguus devised the calendar around AD 525 to work out new tables that would calculate the date of Easter, but the calendar itself would not see more widespread use until the 8th century. The Dionysiac Easter table represented one side in a multi-faceted and intense debate over how to date Easter. The table was only officially adopted in England for Easter calculations at the Synod of Whitby in AD 664, to the consternation of those that preferred other methods. However, more widespread usage in England would not pick up until the eighth century, spreading to the rest of Western Europe over the next few centuries. The salient point is that in the first few centuries after its creation, the table was used primarily for Easter calculations, not as a widespread calendar for ordinary use, even among the ecclesiastical community.
The second reason is that in the years leading up to AD 666, early medieval writers preferred other dating systems when formulating countdowns. These 'countdowns', or summae annorum, drew largely on conventions popularized by Isidore of Seville in his influential universal history, the Chronica maiora. While Isidore employed the anno mundi(AM) system used by authors within the genre, historian James Palmer argues that he was the first to use the six-age conception of human time as an organizational framework within a universal history in combination with the AM system. Like the AM system, the idea that human history was divided into six ages was nothing new, but since Augustine the belief that these ages were comprised of a predictable or fixed number of years had gone out of style. Some of these earlier theories had revolved around the idea that each age would be 1000 years in length, with AM 6000 marking the start of Christ’s millennium-long reign on Earth. According to Isidore, the Incarnation had inaugurated the sixth and final age in AM 5197. His readers in the seventh and eighth centuries picked up on this, and in Ireland and France we see a flurry of countdowns to AM 6000.
I’m going to skate over a lot of the disagreement, but it’s important to point out that the popularity of Isidore does not mean there was consensus on the exact age of the earth. Julian of Toledo famously proclaimed that AM 6000 had passed in AD 675 without event. Julian was reacting to a group of Jews who claimed that the earth was much younger than previously supposed, arguing that Christ was born to soon to have been the messiah if the six 1000-year ages were assumed. Julian rejects the 1000-year ages, adhering to the Augustinian tradition that ages were not fixed and that the End could not be known.
Despite disagreement about the exact age of the earth, Julian’s position reflects one crucial trend that seems to be fairly constant throughout the seventh century: there doesn’t seem to be a strong connection between AM 6000 and the apocalypse— Palmer argues that it wasn’t until the eighth century that there is evidence of attempts to resurrect the connection. Even then, the AM calendar never experienced widespread use in the West. More popular was the use of regnal years. Think, “in the nth year of King [Name].”
All that to say, in the run up to AD 666 it was unlikely that many folks had the AD calendar in the back of their minds ticking away. On a practical level, people kept track of the current year using different frameworks. Additionally, while there was general agreement that Christ’s birth had inaugurated the last age and that the End was imminent, there does not seem to have been any general consensus about when that would be. Most writers still adhered to the Augustinian tradition that the End could not be known. However, when writers did sit down to calculate the current date relative to the steady march of human history, they used a method more conducive to the task.
Sources:
C. Philipp E. Nothaft (2018). Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe.
James T. Palmer (2014). The Apocalypse in the Early Middle Ages.