How do I save my actual SSH key (not passphrase) in the macOS Keychain

I want to store my SSH key in my iCloud keychain so it is available on all my devices. Checking this site and others, every answer to the question “How do I save my SSH key to the keychain” says to use ssh-add -K ~/.ssh/private_key_file, but that does not save the key to the keychain, it saves the passphrase for the key to the keychain. To use the key on another device, you still need to copy over the private key file.

Is there a way to store the actual SSH private key in the keychain so that I can use it without having a key file on my drive?