new cord, and also check to make sure there isn’t any dust/gunk/grit accumulation in the iPhone port. I’ve had failures to sync that were cleared up by cleaning this area (with a flashlight, paperclip and deep breaths).
Also, I still sync my iPhone to my High Sierra Mac (bc that is where my music library lives), so I know it is possible to do despite what Apple Support told you. Patience is required for sure.
If you have one, I recommend a toothpick over a paperclip. Softer and less conductive, but it pulls pocket lint out just as well. I have to clean my iPhone’s Lightning port every six months or so.