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.
Come on we're playing semantics. Yes, obviously a SPV wallet doesn't fully verify like a full node does, but it does simply verify transactions which is the point. It has limited trust to a third party and since the information it receives from a third party is cross verified with others and it is deep in the blockchain, there's a fair assumption of truth in it.
The point is that every person isn't going to run a full node, and you want all users to have a decentralised self verifying experience. SPV nodes on mobile are the best answer to this.