'Chinaman' is an older term and not used much nowadays. Many regard it as derogatory, which is why it has fallen into disuse. There are words, like Spaniard, Finn, Dutchman, that can be used for a person of a nationality, but some are regarded as demeaning and are used less. I very rarely hear it now. Those who do use it are, I think, more likely to be unaware of its connotations than trying to be rude. I read that the British cricket club in Singapore used to have a sign reading "No Dogs & No Chinamen", which would explain why people try to avoid the term nowadays. I don't now whether this is a true story or not, but it is entirely plausible and explains why we avoid it today.
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