Is it possible for me to use these folders with syncthing? I got some advice but it involved me rooting my phone, which seems excessive. Do I have any other options? The Syncthing app is not available for my NAS (Synology).
You could try and mount that on your linux machine and sync to that, but using network filesystems is sometimes flaky (directories becoming unreadable due to connection errors, causing syncthing to think stuff disappeared and broadcasting the deletions to others), so proceed at your own risk, as it might cause data loss.
Also, see if you can switch to NFS oppose to CIFS.
What Synology NAS do you have. If it’s not a really old or really cheap one, you can run Syncthing on it.
Android only needs root, if you want to write to SD Card (if your vendor doesn’t do anything special to circumvent the default Android limitations) outside the Syncthing folder (/storage/.../Android/data/{appname}/ or /storage/.../Android/media/{appname}/)