What is the difficulty with the use of a lunar calendar? It isn't clear when a lunar month begins and ends.

Over time, the months cycle through the seasons.

It didn't help with telling time during the day.

The Moon isn't visible when it's in a new phase.

the old alchemists made significant contributions to science even though there starting premises were off a bit. their conclusions after failure would steer them into better directions for investigation. science is no more than close observation and the initial mistakes result in rethinking and retrying experiments.