The police are not expected to replace the ID. They are expected to give you a proof that you have indeed lost one. In fact Russian embassy won't give you a "returnee permit" unless you go to a Spanish police station and declare your loss of a document.
Even foreign police cannot be expected to just generate documents on a whim.
In the US states don't have embassies, but surely a police station in New York can ask the registration office in California to send them your picture by WhatsApp and have at least a vague kind of proof that you are who you claim to be.
Of course, you can always say that America is exceptional, and she will have "Marxism with American characteristics", just like China switched from true socialism to "socialism with Chinese characteristics", but would still recommend avoiding the word which associates with GULAG and mass starvation.