Jim is correct as it is all to do with cultural context.
It is more to do with the size of the spoon rather than what it is used for.
In recipes, you will have 'cuillère à café' (teaspoon) and 'cuillière à soupe' (tablespoon).
The 'cuillière à soupe' is different in shape from the English equivalent less round but as large.
You can say, petite cuillière and grande cuillière but you will never say -
cuillière à thé
I agree with your logic but I recall having a similar query with my step-son who lives and works in France.
His explanation is that coffee is consumed much more than tea in France. I wasn't impressed with that excuse, but that seems to be the practical explanation.
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