The main issue I’m trying to solve is that when photos are transferred to the iPhone’s "Photos" via iTunes syncing, the picture gets transformed.
To verify that, sync a photo using iTunes to get it into the iPhone's "Photos"; and while iTunes is still running and the phone is connected, access the photo on the iPhone from the computer via USB and check the properties of the photo. It's a transformed lower resolution photo. Called Apple Support about this and they confirmed, that's what it does, works as designed, "to save space" was the reason.
Need a solution: to be able to use a photo organizer that does not rely on media being kept in iPhone "Photos" so I can use a photo organizer that does not transform any of my media.
Best Answer
Some camera apps have a “Lightbox” feature which collects up all the images you capture. When you use the Lightbox to review images you've taken, you can choose to edit the image, save it to the Photo Library, or share it with the standard iOS share sheet, which may include a way to save the image that doesn't alter it the way that @kelly-smith says that iTunes Sharing does.
An example of a camera app with this Lightbox feature is both versions of Camera+, (with whom I have no affiliation).
This also assumes the images come from the phone’s camera, rather that (say) screenshots or downloaded.