I don't remember having a problem with PGR when I finished it in 2002. Although I was playing the 'Classics' re-release (I wonder if the original version had a few bugs in it????) Great game, better than the second one, I thought, because it runs at a much smoother framerate.
Are you playing it on an original XBox, or on a 360? I've been playing Links 2004 on my 360, and whenever I power the console off, all my saved data is deleted, which didn't happen when I used to play it on my old Xbox1.