Battery voltage sag is a result of the chemical reaction and ion migration that takes place inside the battery to supply current to a circuit.
It's like having your headlights on and turning the ignition key to start, the lights dim.
Battery cannot keep up with load demand. Battery has a de facto governor on it.
It can only work so fast.
Dirty battery contacts increase the resistance of the circuit, so the current demand is less, so less sag.
To test this, get out a meter. Test battery voltage. Clip in a resistor. What is the voltage?
Resistors limit current.
To get a better idea of what is going on, you need to look at the battery discharge curves. They are not straight lines - which is what you would see if you were using an AC 'power pack' to drive the circuit, since delivery is not governed by chemistry.