A CA is not a way to share a public key in the first place. The idea behind signing a certificate (which contains the public key) by a CA is to propagate an existing trust into a CA to the certificate. How the signed certificate is then shared does not matter because nothing is secret in it and the recipient can detect any modifications since these would invalidate the signature from the trusted CA.
In practice there are many ways to securely share a public key. The main requirement is that the transport or container can be trusted trusted and that it protects against manipulation of the contents. This can be achieved when communicating with TLS with a trusted server, by sending the key inside an S/MIME mail signed by a trusted sender, by signing a certificate by a trusted CA etc.
Web of trust - e.g. PGP. This is a peer-to-peer system, where instead of CAs that everyone trusts, each user trusts some keys directly (e.g. ones confirmed at a key signing party) and can follow a chain.
Out of band verification - e.g. SSH. When an unknown key is encountered, it's displayed for the user to verify out-of-band (although no-one bothers in practice) and if valid it is stored. Some people also share their known_hosts file over a secure channel.
Secure channel - e.g. Group Policy in an AD environment. The AD administrators can leverage existing secure connections between DCs and other systems, and use this to deploy new certificates.
In most environments, private keys are not rotated frequently; validity would typically be measured in years.
I am not certain you fully understand what a CA is.
A Certificate Authority (CA) is an entity who tops a Public Key Infrastructure (PKI) hierarchy. Its public certificate is the part that needs to be implicitly trusted by all parties in order to enable validation of signatures in a hierarchy.
Public keys are by definition public. The concern is having those intercepted and replaced in transit - in an elaborated scenario, this could completely defeat a PKI should the attacker be able to permanently monitor and intercept all communications.