I figured out my problem at last. The installation notes say to extract the files to the root FF8 directory, but actually it's the Hardcore folder itself that needs to be in the FF8 directory. The installer looks up a directory to find the data folder and when I had it in the FF8 directory, it would go out of the directory, fail to find the data folder, and would exit prematurely and declare it was finished.
So if anyone else is having the same problem, make sure that you have the patch files still inside their own Hardcore folder, inside the FF8 directory.