If the option "Use as System Photo Library" is dimmed, then the library is already your System Photo Library.
How is your external volume :"Photo 3 TB HD" connected? The volume needs to be locally mounted with a wired connection, not accessed over the network. Is the volume directly plugged in?
It seems that Photos can use the Photos Library correctly, if you can launch Photos and it is showing the correct path to the library. Can you view the photos and add adjustments?
What exactly happens, when you try to sync the iPhone?
when syncing iPhone, it goes through all the stages but can't see the photo library as it's grayed out as an option to elect photo library in the syncing page?
Could you post a screenshot of what you are seeing when syncing?