Exactly, great observation. I'm glad you noticed the result of kettlebell training is improvement in kettlebell lifting. I'm waiting on 15mph to post that you need to squat and power clean to get faster. The one thing I will guarantee is that whatever "strength training" you do will most specifically increase you proficiency in that specific list/skill, so choose the lifts based of specificity.
We all know the SAID principle: specific adaptations from imposed demands. So get specific with it. Ask questions any collegiate coach would ask during periodization planning: what is the overall goal? What qualities are necessary to perform the skill? What is the distance/time required to perform or complete the skill? How can I add a resistance or strength element towards this specific motion/ skill.
Stay specific and you can't go wrong.