It's faster than `ct<somechar>` for replacing part of a word.
Using `<Insert>` to toggle the mode is also something I do fairly regularly ...
I'm past the point of consciously doing these things so I probably use it more than this.
So for example this morning I was editing cells in a latex array. They are delimited by "&" and each one is some sort of equation that doesn't fit a normal vim word definition. R to the rescue. If I need more characters than are there in the cell then I go <esc>I to carry on but this morning, R was perfect.
I never toggle insert mode with an ins key because I got in the habit of using minimalist keyboards with weird custom mappings that didn't have that key (or not that readily available) often.