The issue is if you are emailing a counterparty with whom you have no trust established yet, there is no way for them to trust your public key (or for you to trust their public key) unless you have somehow exchanged keys in a secure manner. This is the key distribution problem.

Certificate authorities help manage this by notarizing your public key with theirs and if the CA is trusted by the counterparty, they can use it to validate the public key is from you.

Having said that, it would be great if mobile operating systems had services that could support key notarization and if you could configure say 4-5 of your trusted contacts to act as notaries and they vouch for your public key automatically. With DID-core, this may become possible.