Trade Balances and Capital Flows

Looking for experts on international trade and finance out there...I have never been able to fully grasp the basic textbook macroeconomic accounting identity stipulating that trade deficits = capital inflows

For example, the classic example always presented is that if the U.S. buys $100 worth of Chinese goods (imports/trade deficit), the Chinese manufacturer then turns around and purchases $100 of U.S. treasuries (capital inflow) .

Doesn't this logic fall apart if the Chinese manufacturer decides to store the $100 in the bank, or under the mattress? What if he simply lights the $100 on fire? In any of these scenarios, how does the trade deficit = capital infow hold true?

This may sound like silly/immaterial question, but I suspect my inability to mentally reconcile something seemingly so trivial limits a deeper understanding of the dynamics of internal trade and finance.

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