Given that we can record how much fuel was put in, and how fast the motor was spinning at each point in time, why can't you model how much fuel is left in the tank? Is fuel consumption-to-motor-use very variable?
It's up to the Chief Engineer if he/she wants to use an Excel spreadsheet to track fuel consumption (and most probably do). As for tracking it based on weight, maybe some chiefs do that, but I imagine it's not worth it for most.
I wasn't a chief, so I don't know what their responsibilities are in regards to fuel consumption. But I imagine they would have to give estimates of future fuel needs to the company based on the ship's schedule.