It is exactly as you suspect. It is totally illogical to consider , for example, to be three different words and here is why.
Whenever you are dealing with a kun-reading word, you need to remind yourself that it existed when Japanese was merely a spoken language. We had no way of writing or even . All we had was the sounds "katai". Then we encountered the Chinese and the rest is history.
The vast majority of homonyms in Japanese are found in words of Chinese origin.