SPV does verify. I don't know what definition you're using when you say it doesn't self-verify, but it does verify transactions itself and doesn't rely on a trusted third party to verify.
That is untrue. A SPV wallet cannot fully verify a transaction. It does not have the UTXO set, so it cannot fully verify it. Even if it does know the previous transactions, it does not know the transactions that those spend from and so on. If it did, it would basically be a full node, which they are not. Because an SPV node does not have the UTXO set and has not verified the chain of transactions that a given transaction is part of, it must trust the node that gave it the transaction and trust that that node gave it the right transaction and that the transaction was valid to it. Without the previous transactions, it cannot verify that the input script is correct. Without the previous transactions, it cannot verify that the amounts in the outputs are valid.