This article has some information about how Let's Encrypt enabled an "automated process that handles renewals": https://duo.com/decipher/proposal-to-make-https-certificate-...

I wonder if such a process should be made an industry standard? Does anyone know if there are any proposals for it?

Let's Encrypt literally is an implementation of the industry standard; the standard is called Automatic Certificate Management Environment.

But, somewhat annoyingly, it is only seen as applicable to the public internet. There's no effort to make ACME based CAs for non-internet usage.

For internal use, create and distribute you own root CA with self-signed certificates.

And by letting smallstep/certificates [1] handle ACME, it's just as easy as using LetsEncrypt for public certificates.