As this is asked on Information Security, it's crucial to first concentrate on how the keys should be trusted. Unlike SSL PKI with certificate authorities (CA), PGP is based on a web of trust. Anyone could send you an email with a self signed PGP public key, claiming it's really the key of the sender. It's as easy to spoof as the email itself. The responsibility of accepting and trusting it is on recipients side. Building such feature to email clients would create false sense of security.
Retrieving keys directly from key servers sounds like a bit better idea, but it doesn't actually give any guarantee, either. Anyone can send keys and signatures to the key servers and it's better to manually check who else trusts the unknown key, unless you can't check from independent source that the fingerprint matches. The keys would be retrieved using a keyring management software instead of the email client; e.g. the Kleopatra certificate manager and GUI for GnuPG a built-in functionality for this.
Best method for validating a key would be to ask personally, but I'd publish the key on a CA signed TLS protected web site, possibly with a DNSSEC signed IP address. This way your key would be behind two independent public key infrastructures (SSL + DNSSEC), making the source quite trustworthy regardless who have signed your key and could the recipient trust your web of trust.
For example, many national computer emergency response teams (CERT) publish their PGP keys on their HTTPS websites, and some are properly signed with DNSSEC, too:
There is an experimental RFC 7929 for DNS-Based Authentication of Named Entities (DANE) Bindings for OpenPGP from 2016, introducing a new DNS Resource Record (RR) type,
OPENPGPKEY. This would be easy to automate on an email client. However, the RFC 7929 itself notice the problems with the security of this approach (section 7.6):
Anyone who can obtain a DNSSEC private key of a domain name via coercion, theft, or brute-force calculations, can replace any
OPENPGPKEY record in that zone and all of the delegated child zones. Any future messages encrypted with the malicious OpenPGP key could then be read.
Therefore, an OpenPGP key obtained via a DNSSEC-validated
OPENPGPKEY record can only be trusted as much as the DNS domain can
be trusted, and is no substitute for in-person OpenPGP key
verification or additional OpenPGP verification via "web of trust"
signatures present on the OpenPGP in question.