The reason why you have to adjust for zero on analog ohmmeters is related with the way they measure resistances. Below we have a simplified schematic.

For you to measure R
U, you need to know V
B and R
K. However, batteries change their voltages as they discharge, so they are unreliable references. To mitigate that you have to calibrate your meter to the actual voltage. You do that by shorting out the leads so that V
B = V
R. Then you adjust to zero ohms which is in fact the full scale of the galvanometer.

R
K changes with the range, which means a different load. Since the battery has internal resistance, it will show a different voltage for every range. That's why you always have to adjust for zero every time you change the range. This ends up taking into account the resistance of the leads, but that's not the main reason why they need that calibration.
Digital meters on the other hand use precision reference currents and voltages, and perform automatic zeroing for each sample, so you don't need any kind of manual calibration, just an occasional compensation for the resistance of the leads.