Let's remember that Kirk's historical knowledge seems to be very narrow: 19th century United States, exclusively. He's lost at sea in early 20th let alone late 20th century United States, and has never demonstrated knowledge of other geographic areas of Earth in his own era let alone any historical ones.
This would be understandable if Kirk lives in the here and now, and has studied a very specific incident of "ancient" history plus its immediate surroundings as a hobby. Perhaps he knows about gunslingers because he's a fan of Lincoln, or vice versa.
Now, had TOS indeed taken place at one of the "alternate" timepoints offered - a few more centuries in the future, from where we're looking - his narrow focus would probably mean he would not
know about spectacles or lightbulbs or typewriters, as these would be quite peripheral to his interests. And he'd know basically nothing about tommy-guns or cell phones, as that stretch of history would never have been his hobby, yet it would have been such distant past that not even his granddaddy would ever have told him stories or shown him pictures.
It's the mere 250-year gap between the last spectacles we see and Kirk's days that makes it all so difficult to swallow. But perhaps people in the future are not particularly history-conscious overall? I mean, we might all plausibly have seen photographs of the walking aids polio victims had to wear, but in general those are far outside our experience, and we'd certainly need instructions to make use of those.
In the end, it's just as well that this is idle speculation and the news turned out to be false. But Kirk should still never be mistaken for somebody who knows his Earth history!