Looking at it as “there is a kanji and it changes in different words” is the wrong way to look at it. There are words, and you can write the words with kanji. The words exist, and then kanji that match the core concept were used to write the words.
All the words can be written without their kanji as well, but then it becomes harder to tell words written identically apart. Or, for beginners, it can be hard to tell where words end and begin.
Many words are also borrowed directly from Chinese.
But thinking of it as words first and then ways to write the words should help. Going in the other direction, yes, it would be kind of wonky.