Clocks have discrete ticks. They are digital devices. Even a base-60 second hand is digital because the number of states is finite.
Mechanical and digital are not mutually exclusive concepts. For example, "The analytical engine was a proposed digital mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage." -- https://en.wikipedia.org/wiki/Analytical_engine
Going further, I could argue that the digital age is very old. Humans who wrote numbers for accounting purposes were engaging in a digital activity; only the numbers matter, not the medium they were written on or the exact handwriting style of the scribe who wrote those numbers. DNA is a form of digital data conveyed through a sequence of 4 possible symbols, and DNA predates humans by billions of years.
The pedantic phrase substitution for "pre-digital age" would be something like "age before widespread digital electronic computers on solid-state microchips" (thus differentiating from analog electronic computers and vacuum tubes).