I understand there's probably a big thorny problem of duplicating the corporate process/policies on the human level that ensure compliance, but is the back-end software pipelining stuff to CT logs not also something that can be replicated? Or is it not part of the server side stuff which has been open sourced?
Our code for sending stuff to CT logs is fully open source. But that's the tiniest slice of our compliance regime -- the vast majority of it is things like audit logging certain events, preserving audit logs in specific ways for certain amounts of time, ensuring dual-controls on all systems, being both audited and penetration tested annually, maintaining firewalls and vulnerability scanning tools, etc.
It's absolutely possible to spin up another new CA; lots of folks have done so over the years. But having time, and money, and prior experience all help a lot.
https://letsencrypt.org/docs/ct-logs/