The different colours of chromatic dragon were inspired by more than just damage types
The Slayer's Guide to Dragons (2002) includes a preface from author Gary Gygax which explains, to some extent, his thought processes when designing the original five types of chromatic dragon in OD&D.
The first type of dragon Gygax designed, and used in play, was what would become the typical red dragon; a fire-breathing winged reptile, the model for which was a converted dinosaur:
As a matter of fact, what was probably the first dragon used on the table top was a converted dinosaur model [...] with the addition of cardboard wings, the general form of the fearsome red dragon was visible!
For the sake of variety he then added poison-breathing green dragons, inspired by some legends that described dragons as serpents with poisonous breath as opposed to breathing fire, and some that specifically mentioned green dragons (it's unclear if they are the same legends):
Some 'historical' references spoke of dragons as 'serpents' with poisonous breath. There were mentions of dragons of green colour. Thus, it was a simple matter to add the green dragon that exhaled a cloud of poisonous gas [...]
White dragons were next, apparently because that would complete the triad of red, green, and white dragons found in mahjong tiles, and the cold nature of their breath was chosen to match their colour:
[...] the mahjongg game has three sets of different tiles named dragons - green, red, and white. Having played that game since I was a boy, how could I ignore the white dragon? So what form of breath weapon went with that colour? Snow and cold, of course. [...]
Then blue and black dragons were added, and it seems that this was indeed to represent the extra damage types and add even more variety to the kinds of dragon available for players to fight:
After some contemplation, I added the blue colour, as that could well represent lightning, and there was a spell in the rules covering just such an electrical bolt. Acid breath seemed another reasonable form of attack, black represented that well, and thus the fifth kind of malign race of dragons was born.
That ended the near-complacency of would-be dragon slayers. No longer could a single set of defences and attacks apply when a dragon was known to be on the loose.
So according to this book, red dragons exist because of the common depiction of the fire breathing dragon in European myth, green dragons exist because of myths about poison-breathing serpents, white dragons exist because of mahjong tiles, and blue and black dragons exist for damage variety.