Thank you for trying this. This confirms that the missing publisher logo error you were getting is as a result of a plugin conflict with Rank Math SEO.
I have tested the plugin myself with the same result and have checked to see if there are options to add the publisher logo manually from the plugin settings but this does not look possible. I have also checked to see if you can disable Rank Math‘s schema capabilities for stories but this also doesn’t look possible.
As for the missing author url messages you are now getting I presume you are getting these once the Rank Math SEO plugin is disabled? The Author URL is an optional field in structured data and not mandatory.
In this case I would contact the Rank Math team regarding the missing publisher logo error to see if there is something they can do to fix that. Then if they can the optional warnings you have should no longer show if you use the Rank Math plugin.
In the meantime the only way you will be able to avoid the missing publisher logo error message would be to keep the Rank Math SEO plugin disabled.
Hope this all helps and do let us know if you need additional support from us based upon their reply.