Think of the certificate as an official signed statement confirming that the public key belongs to a particular host. The CRL/OCSP systems are there so that you can confirm that the official signed statement is still valid and that is still applies.
In reality, most browsers DO NOT actually check CRLs or use OCSP to check validity because that would be too slow. OCSP stapling does exist, but, as of now, it hasn't really taken off.
One of the reasons to "bother" with certificates is because you need to know WHOM to contact to confirm that the certificate is still valid. Without the certificate, you wouldn't know whom to contact or who the CA is. Another reason is so that there is a signed proof by the CA confirming that they did some checks to confirm ownership, etc. The certificate can contain the name and additional info of the organization, etc. The certificate also contains Certificate Transparency proofs, which are there to show that the CA published the fact that they had issued the certificate to a public log, and this serves to make sure that CAs don't misissue certificates, among other things, it keeps CAs in check, etc.
OCSP and CRLs don't tell you that "this is the public key of the server", instead, they tell you that you can trust whatever public key and other info is included in the certificate.
In general, the host using a certificate is supposed to keep the private key safe and there is the assumption that they will keep it safe.
Another issue with OCSP when the client makes the request to the CA is that it does not preserve privacy. The CA knows which (IP address) visits which host (website, domain). Also, if every client makes OCSP requests, the servers of the CA would receive significant traffic and require infrastructure upgrades to handle it.
The best and most practical scenario would be for the certificate (with Certificate Transparency information) to be accompanied by an OCSP response. The server preemptively and frequently asks for an OCSP response to have on hand and gives sends it to the client, along with the certificate. The same OCSP response can be sent to many clients, as long as it is still temporally valid. Also, since the server makes the OCSP request, the privacy of the client is preserved. The loading time is not increased, in any significant degree, since the client doesn't have to contact the CA for an OCSP response (or search through a huge CRL) and the cryptographic verification of the server-supplied OCSP response is quick. This is called OCSP Stapling; unfortunately, it hasn't taken off as of now.