What I don't understand is why the problem is not expressed this way from the beginning.

I don't know for sure - but perhaps a memetic analaysis of paradoxes might throw light on the issue:

Famous paradoxes are often the ones that cause the most confusion and discussion. Debates and arguments make for good fun and drama - and so are copied around by the participants. If you think about it that way, finding a "paradox" that is confusingly expressed may not be such a surprise.

Another example would be: why does the mirror reverse left and right but not up and down?

There, the wrong way of looking at the problem seems to be built into the question.

( Feynman's answer ).