The SPV system is not something that "keeps miners in check". The SPV system is a cryptographically secure way to know that a given transaction is part of a given block chain.
I never said that SPV was to "keep miners in check". You are completely misunderstanding me.
Fraud proofs are necessary to have a cryptogrpahically secure way to know that a transaction is part of a given blockchain AND that the transaction is valid. Yes, merkle trees ensure that a transaction is part of the blockchain. But nothing currently exist to prove that a transaction is valid without having to have the full transaction history. The only way that a transaction can be fully validated is to know the transactions that it spends from, and then the transactions those spend from, etc.
In that respect, it is working, and it is working correctly. Wallets like electrum work that way as far as I understand.
No, it does not currently work, and it is not how Electrum works at all.
All that Electrum can do is know for certain that a transaction is included in a block. It must trust that the Electrum servers that it has connected to have actually verified the transaction. However if your Electrum wallet were to be connected to malicious Electrum servers, they could serve you invalid transactions which you would not know are invalid. Said transaction can be included as part of a block; the merkle root would be correct and the PoW of the block would be valid. BUT the block would contain an invalid transaction. For full nodes, this block would be entirely invalid and discarded. But we are talking about malicious Electrum servers here. So those malicious servers TELL YOU that the invalid transaction is actually valid, and so you accept it. There is no way for you to prove that the transaction is valid or invalid, Electrum simply does not have the data to fully verify the transaction. But we still have met all of the criteria that you wanted: the transaction is included in the merkle root and the block's PoW is valid. The big thing that you are missing is that the block includes an invalid transaction, and SPV wallets have no way of knowing whether the transaction is valid or not. Fraud proofs are required to prove that all of the transactions in a block are valid, and currently they do not exist nor is there a known way to make such proofs.
Just because a block has a valid PoW does not mean that all transactions in the block are valid. Just because they are included in the merkle root does not mean that all transactions in the block are valid. There is more to a valid block than just the merkle root and the PoW.
Edit: It's not worth my time to argue this with you. You clearly don't understand how Bitcoin or SPV wallets work. To my ignore list you go.