It is productive. It can't pluralize anything, but it can pluralize anything that refers to people and it is actively used in novel ways. I've seen someone refer to 美国的妈妈们; metonymy is not involved there. It just means "American mothers", as distinct from a hypothetical 美国的妈妈 "America's mother".
However, the other angle on this is that Mandarin pronouns have singular and plural forms (plurals using 们), and the use of the correct form is obligatory, which suffices to show that plurality exists in the language. Although it isn't the case that 们 is unproductive, even if it was unproductive that still wouldn't show that the language has no plurals.
> in this sense it's more of a metonymic marker rather than a plural marker). For example, 白宮们 can be used to translate "the White House" when it refers to the President and his administration, and cannot be used to mean "white houses"
I should note that this argument doesn't entirely hang together. You can make "the White House" explicitly plural in English by giving it a plural verb:
https://us.iasservices.org.uk/bidens-immigration-bill-propos...
> The White House have announced a comprehensive immigration reform proposal in a bill that has been sent to congress.
How would you say that differs from 白宫们? Does it refer to multiple houses?
When the comment you replied to mentioned "Chinese has gotten by for thousands of years without any plurals at all", I understood it to mean that Chinese has not featured any general system of marking plural by grammatical means[1], which is what is usually understood by the term "plural"[2], not that Chinese has no ability to express a more-than-one count distinction at all (which isn't the case in any language as far as I'm aware).
> It can't pluralize anything, but it can pluralize anything that refers to people and it is actively used in novel ways. I've seen someone refer to 美国的妈妈们; metonymy is not involved there.
It is productive in a limited sense in that way, but not as a general plural marker as you're arguing, and it's limited because 美国的妈妈们 means "American mothers" in that it necessarily refers to them as a collective group (which I argue is an instance of metonymy) rather than a set of more than one "American mother". For instance you cannot say *三个美国的妈妈们 to mean "three American mothers"; you must instead say 三个美国的妈妈 because 美国的妈妈们 can only ever refer to the entire collective group.
> I should note that this argument doesn't entirely hang together. You can make "the White House" explicitly plural in English by giving it a plural verb
This is a feature of UK English where collective nouns agree with plural forms of verbs. US English on the other hand, requires the singular form[3][4]. This has no bearing on how we analyze Chinese.
[1] https://en.wikipedia.org/wiki/Grammatical_number
[2] https://en.wikipedia.org/wiki/Plural
[3] https://victoryediting.com/collective-nouns/
[4] https://en.wikipedia.org/wiki/Collective_noun#Examples_of_me...