I've seen many threads that contain bits and pieces of the solution, but here is what I did. Ignore the whole uninstall/reinstall/firewall bs. At least I've been through all that to no avail. Here's how I got this solved:
ON YOUR MACBOOK: So you have your local files that appear on Spotify. Make a new playlist and call it "Local Files". Drag all the songs (or whichever songs you want synced to your iPhone) from the original Local Files on the Spotify sidebar, to your new Local Files playlist.
ON YOUR iPHONE You will see the new "Local Files" playlist appear. You click download, but the songs are not syncing when you try to download the playlist. The playlist is either "waiting to download" or "waiting for wifi" and most of the songs are greyed out (minus the tracks that Spotify found in its own database--those may be downloaded.) However, keep the playlist switched to Download.
HERE'S THE FIRST STEP THAT MOST "SOLUTIONS" ARE MISSING: On your iPhone Spotify app, click "Your Library" in the bottom right corner. Now on the top right, click the Settings icon. From there, click on "Streaming Quality". Scroll to the bottom and you will see "DOWNLOAD USING CELLULAR", meaning you will be able to download songs over 3G/4G/LTE. MAKE SURE THAT IS TURNED ON. After it is turned on, go back to Your Library, and exit the app.
Create a personal hotspot from your iPhone. Settings -> Personal Hotspot. Turn that on, and join that hotspot on your computer under the WiFi icon on the top bar. Keep Spotify open on your desktop. Open Spotify back up on your iPhone.
Go to playlists on your iPhone. If you haven't yet, download your new Local Files playlist.
This, at least in my situation, magically cured this issue. All of the files now started downloading on my iPhone.
After success, be sure to turn off the "download using cellular" unless you don't mind Spotify downloading music when you're not on wifi.
Let me know if this worked for you!