Why can Image Capture not transfer photos I can still see on my iPhone?

Image Capture can only see photos in the Camera Roll.

It will not see photos in iCloud, if you are using iCloud Photo Library on your iPhone, and it will not see photos that have been synced to the iPhone using iTunes Photo Sync. Did you use iTunes to sync the photos to the iPhone, or do you have iCloud Photo Library enabled?

I cannot test this right now, because I do not have my iPhone ready. But if you select a photo in a shared album you can save it to the Camera roll. Try the "Share" options.

Or you could enable iCloud Photo Sharing on your Mac as well. If you can receive the shared photos on your iPhone, you should be able to se the sane shared albums in Photos on your Mac, if you enable iCloud Photo Sharing in Photos.