So I dug out our old iMac - this version from summer 2000: http://apple-history.com/imac2000
Although I can no longer run TR 1-3 on it I was miraculously able to run TLR on it. However, I can't save! I realized that I couldn't save while I was playing the training level so I thought that maybe you aren't supposed to be able to save during the training, but it carried through to the actual game. Now that I think about it, I vaguely remember having this same problem when I played the game for the first time years ago.
Basically every time I try to save the save slots never change from saying "Empty slot" to saying the level name. When I attempt to load what I just attempted to save Laura says "no". I know it's not an issue of having to find "save crystals" or anything because the Mac games never rely on save crystals; you can save whenever you please.