You can look at the INS position (and compare the position from each INS separately), but practically you're always looking at the "FMS position" which is where the flight management computer thinks you are. That takes INS, GNSS/GPS and other nav beacons into account. So it's technically true that pilots don't often look at the specific screen that has the INS position(s) on it, but you are looking at the FMS position which is basically equal to the INS position if GPS is not updating and you receive no ground signals.

Modern aircraft will also set the known position based on the runway you're taking off from. And when airborne they'll pick up a DME/DME fix and update the location. So while it's procedure to set the known starting location at the gate, there are also other sources.