Why do so many people think you can verify that outputs are unspent using SPV methods!? Yes, you can verify that a transaction has been confirmed; but no, you cannot verify that an output of a transaction is unspent.
Satoshi COULD HAVE made this possible. Miners would simply commit to a root hash of the UTXO set each block, similar to how they commit to the Merkle root.
My guess is that Satoshi didn't know how to implement this. Given all the other work on his plate in designing and building bitcoin, he never took the time to figure it out.
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