As a gamer, I'm drawn mostly to exploration and action. So linear, puzzle-based adventure games (achievement-focused) aren't the sort of games I play often. But I do have some experience with puzzle-based gameplay.
Puzzles are great, but the reward for solving them doesn't always have to be narrative/game progression. Like novels involving mystery, games involving puzzles could be completed without players being required to solve all or even (perhaps) any puzzles.
Instead, solving the puzzles could offer the player a more accurate or complete picture of the story. And, like characters in popular novels, the player might have only one opportunity to notice/find clues or solve a puzzle before the story moves forward. Keep in mind... I'm not trying to imagine a particular game system here so much as raise questions about the use of puzzles and clues in games.
The original Neverwinter Nights included a mission in which the player acts as judge over two brothers. The player investigates a crime by speaking to all parties involved, then decides who will be punished. What I love about that mission is that, if my memory is correct, the player can pass judgement and the game moves forward regardless of whether or not the investigation is thorough. You can judge wrongly (not just wickedly, but mistakenly) and keep playing.
That mission did not affect the larger narrative, but a similar scenario could be used to offer the player insights into the central plotline, side narratives, characters, setting, or any other story element. The player needs to be assured some things to feel satisfied at all when the game's end is reached. But, if replayability is supplied by non-narrative elements (like action or exploration), then the player can be allowed to reach the end without having the fullest or clearest picture of events.
If well done, the ending would hint at what replaying could offer: where the gaps are in the player's understanding of characters and events, what unfortunate events could be prevented or rectified, etc. The original Doom game told the player after each level what percentage of secrets he or she discovered, forcing the player to wonder what was missed. Deeper games could accomplish the same effect in more artful ways.