Most classical isn't designed to help you think. It's designed to be evocative and focus your attention on the music. Classical music, played well, is hard to ignore. This is something that's been lost - orchestral music is event music, ceremonial music, music to fill your world. Steve Roach, on the other hand, can just sort of play for literally days.
Composers definitely have intentions when writing a piece and, if they're any good at all, some of those thoughts and feelings are conveyed to the listener. However, I think the issue is less about what the music was designed to do and more about people's general perception of music. Here's two things that I've observed and believe to be true usually: 1. Repetitive music is harder to ignore. 2. Music with words in your language is harder to ignore. If I put on music to concentrate on something, it would need to be music that can be temporarily ignored and then refocused on when I finish the micro-task at hand. Rinse and repeat and those moments eventually build up to your finished essay or math homework. I'd say this is why concentration music, for me, is almost always classical or jazz. Don't misunderstand me, I have a deep affection for both of those genres and have been moved deeply by certain tunes from both of them. But that doesn't stop me from admitting that often they work equally well as background.
It's funny you say that because when I listen to classical music to help me study I definitely am paying attention to the music but it's still has the effect of quieting the mind enough to then concentrate on something else like studying. The same thing happens with cleaning, I stop thinking of all the other tasks and focus instead of wanting to jump to a different task.