I think the biggest issue come playoff time is lingering injury issues, rather than fatigue.
The reason we lost the Boston series is because in Games 1 & 2, no one played. It's not because they were tired, it's because they just didn't show up.
However that said, a goalie who plays 66 games, likely doesn't have the time to recover from any lingering injury issues before entering a (hopefully) 20+ game playoff schedule.
Hence, you need a backup, who down the stretch can allow the starter to play, maybe 60% of the games, to end up around 55-58 on the season, and enough time in the last two weeks of the season to get rid of any lingering issues.