log (n.2) "record of observations, readings, etc.," 1842, sailor's shortening of log-book "daily record of a ship's speed, progress, etc." (1670s), from log (n.1). The book so called because a wooden float at the end of a line was cast out to measure a ship's speed. General sense by 1913.*
So it does have a relationship to the wooden log - it was originally a series of measurements from a floating bit of wood.
Also, log can refer to the gamma curve which affects how roll-offs in highlights/shadows are saved (rough description).
"Log" is still used daily in production work.
To get to 2nd AC, you had a good sized crew
The fact that a "log" is still a thing in today's productions is not dependant on which crew member is doing it. In fact, with all of the digital f/x, they are crucial.
https://www.mvd.usace.army.mil/Portals/52/docs/regional_floo...
Film was also sold by the foot. You didn't by film by 15 minutes. You bought film by 1000' rolls (35mm).
As ingenuous as our mechanical past was, I am so happy to have lived through the transition to 100% solid state everything.
But that just makes sense doesn't it? It's only 15 minutes' worth if you run it at 1000'/15m.
Developing your film was also charged by the foot. The soup didn't care about your framerate. Just how much was need to process.