I tried uninstalling the plugin from Chrome and then reinstalling it. It seems to have fixed the problem that I got when I would click on the "View all files in Google Drive" button. It now correctly pulls up the Paperpile folder in Google Drive.
It also made it worse for accessing files in the Paperpile web viewer. Before doing this, I was able to view PDFs in the Paperpile web viewer even if files didn't show up in my Google Drive. Now I can only view files that are available in my Google Drive. If I click on "View PDF" for the files that did not get synced into my Google Drive, I just get an error saying "Something went wrong – could not open file."
I tried forcing a sync and Paperpile tells me that everything is synced. The files are still not in my Google Drive.
I tried trashing the reference I mentioned previously and then going back to Google Scholar to add it. It found the PDF correctly and synchronized with Google Drive now.
It seems like unless Paperpile can figure out what went wrong, the only fix is to trash references that aren't synchronizing individually (make sure to download all the PDFs if you have annotations or you will lose them) and then go back and re-import them. Extremely tedious and time consuming, but might be the only solution.