Conditional Fields CF7 plugin not working with More Mail

This is not something I can address. Supporting more than the default two emails is completely up to the author of the Conditional Fields for CF7.

After a cursory review of that plugin’s code, I see no use of the wpcf7_additional_mail hook which could alert it to the presence of other messages, and it appears to have the expected mail properties hard-coded.

The thing is that any plugin can use that hook to send additional mail along with a form submission. It may not be easy it for Conditional Fields to apply its extra logic onto such messages, or even know whether it should.

You may need to consider writing custom code, e.g. your own plugin, as whether to send mail isn’t controlled by just deleting the To: address of an email. You should be removing unnecessary messages completely by attaching to that wpcf7_additional_mail hook.