This is a great comparison. We're arguing about the definition of "word", and attempting to expand it to include edge cases where two words with separate meanings have a different atomic meaning when combined.

We could have a similar debate about whether common suffixes and prefixes should be regarded as individual words.

Much like "planets" don't really exist as a separate natural object, words don't really exist in natural languages. They are artificial concepts, and therefore we will always have edge cases.

I would argue that it is still a useful discussion, as it sheds light on the nature of language (or of celestial bodies), even if the definitions defy the same rigour as mathematical concepts.