For example, if weather presenters said 10mm of rain had fallen it would be quite possible and reasonable to mishear that as 10cm...A good reason for not using cm is the potential for confusion. For example, if weather presenters said 10cm of rain had fallen it would be quite possible and reasonable to mishear that as 10mm, the error isn't immediately obvious because a factor of 10 difference is within expectations. If everyone sticks to mm or metres then the difference factor is 1000 and an error should be obvious just from the numbers. If we get 10 metres of rain we ought to be very worried.
Actually, as they're measuring a liquid, you might even think they said 10ml