Kicking does indeed have a role in helping you propel your body forward, but I would argue that for most people, the most important thing it helps you with is to give you a stable platform for the movements of your upper body and torso.
If you think about any 50m sprinter in the Olympics, they swim at very high stroke rates and they kick incredibly hard and fast. Without a strong kick they would simply not have the ability to rotate back and forth with such aggression and precision.
The novice swimmer, trying to breathe to one side as she swims in full stroke, needs to trust that when she rotates to one side, she will be able to rotate far enough to find air.
Swim 100-200m, depending on your skill level, without any kicking whatsoever. Have someone watch you to make sure you are not cheating!
Swim 100-200m with a relaxed but high tempo flutter kick, with your tempo per a fast "one-two! one-two!" (or 0.25 seconds per kick). This doesn't mean you're kicking HARD, just fast. Keep your knees and ankles floppy.
Swim 100-200m with fins at the same fast tempo. Same fast and floppy kick.
Swim 100-200m with no kick again.
Repeat this twic
On the second round, pay closer attention to your arm recovery and shoulders when you don't have any kick at all. Do your arms, hands and shoulders feel relaxed? (If they do you may have slowed down too much)
Without an effective kick to help you get into a rotated body position, you will again end up in a flatter swimming position and have to resort to some kind of swinging or chicken wing recovery, which, as we saw in part II of this article series, is bad news.