Consider cat. It's short for concatenate. It concatenates the files based to it as arguments and writes them to stdout, that may or may not be redirected to a file. If it didn't pass along terminal escapes, it would fail at its job of accurate concatenation.
Now I don't mean to dismiss your idea, I do think you are on the right track. The question is just how to do this cleanly given the very entrenched assumptions that lead us where we are.
This is usually knowable.
It's a different question whether cat should be doing that, though – it's an extremely low level tool. What's wrong with `less`? (Other than the fact that some Docker images seem to not include it, which is pretty annoying and raises the question as to whether `docker exec` should be filtering escape sequences...)