When we apply pulse-width modulation (PWM) on controlling the brightness of a LED or the speed of a DC motor, is it the lag in our eyes that is responsible for the feeling of a continuous powering or the inertia of the devices?
Depends on what you are controlling. For DC-motors it is the inertia of the device that acts as a low-pass filter of the PWM modulated signal resulting in a continuous motion. For most LEDs it is the human eyes that do the apparent low-pass filtering.
If the PWM-frequency is not very high you can actually see this by moving your head from left to right while looking at the PWM modulated LED. The light will appear on and off while the LED is moving through your field of vision (this can get very annoying with LED car taillights).
For LEDs the filtering can be done by the eye.
For an electric motor there are 2 sources of smoothing. The first is due to a motor being an inductive load. In an inductor the applied voltage is proportional to the rate of change of the current. This means a PWM voltage will result in some sort of triangle wave in the current (which requires a freewheeling diode for when the switch is off). Motor torque is generally proportional to motor current, so you may actually be applying a continuous torque with some ripply. Then the inertia of the motor helps smooth the speed even in the presence of a non-uniform torque.
You can also reduce the actual flicker of an LED by putting an inductor in series with it with a freewheeling diode but this is not usually practical at the frequency one might PWM an LED.
It may also be informative to read about buck converters - particularly in continuous conduction mode. One can think of a DC motor as a speed dependent voltage source, which will allow you to consider it a buck converter that can transfer energy in either direction if it's moving - i.e. regenerative braking.