Your impression is a little wrong, see below.
> Never do this. Just don't. What you should do instead, is write a game, and while writing that game, write its engine.
I'm not making a game. We have an OpenGL Game Engine class in my Game Programming program at college. I'm creating my game engine with the idea that someone could simply include it as a library and then use its features to develop a game. This Game Engine class is now over, however, I'm still developing my engine purely for personal learning purposes. Yes, I am creating demos to test features of my engine, but I'm not writing anything that would be considered a game.
Regarding the rest of your comment, I totally agree and understand what you are saying. This actually makes sense, however, it's just not the way that we've been taught so far in my program. Being a student, we are familiar with the practices that our teachers use. This makes us somewhat close minded, but it's really nice when people (like you) give a completely different way of doing something.
Thanks for the help! I'll look into these different methods of structuring my engine.