The technical biomechanics in swimming is still an evolving art & science but it is also pretty advanced nowadays. Since you seem very analytical on your endeavour, I suggest that you continue to research in depth as well as get your coach to break down the mechanics of the kick specifically for you and spot train you on how to fix it. The coach should see exactly what you do right or wrong and let you know.
Here is a helpful guide from Garry Hall, on leg propulsion
If your technique is not yet helpful, or somewhat wrong, repeating it over and over on a kickboard may not help you improve. It might even cause you to build physical habits that become difficult to break at a later time. Stronger alone will not make you faster. Stronger techniques will surely do. So, get it right first, then train on it.
Leg propulsion differ a lot from person to person. This is true for beginners as well as world champions. Because swimming is a full body activity, the propulsive power balance through out the body can vary significantly to favor each person's strength. That said, the legs tiring in your case might not all be due to lack of strength. It is possible that you might be kicking from the knees & things too much and not yet allowing hip swing to help in your kicks.
The kicks should be hip and core driven with your legs being delivery extensions. If you tighten up throughout the motion, you will surely be tired twice as fast. You may want to do an easy test of just kicking about 10 kicks as you normally do and see how far. Come back to the wall, focus on hip driven kicks for the same 10 and see if you see a difference. Also try to observe some good swimmers at the pool to learn visually and make adjustments, measure and improve that way.
Swimming is both simple and easy to enjoy, but can also be extremely complex when you dwell deep into the interplay between biomechanics and physics. It's a fascinating science that is executed artfully through the body and senses as it moves forward in a balancing act between propulsion and drag.