I was thinking specifically with MEW in mind, where you input the private key/seed phrase into the "View Wallet Info" tab. With this, you know your private key -> public key without having to send a transaction.

if I generate any public key

Sorry if it was just bad wording, but this is incorrect. You generate a private key, and the private key's Keccak-256 hash (last 20 bytes) is what ends up being your public key. You do need to make sure that the private key accesses the same public key for every instance of storing your private key, not because the math won't work, but because the private key could've been corrupted or copied wrong.