Would it be possible to play through Pandemic Legacy more than once?
Yes, it is possible. Someone on BoardGameGeek has designed a Reset Kit, which gives full instructions for keeping various components separate, rules reminder cards so you don't have to sticker the rule book, and other stickers to go over the ones on the board/cards to set them up afresh.

If you're really careful, you can remove the stickers from the board and have something that is 99% vanilla Pandemic, which is also an option.

However, for many people the permanence of the game is part of the experience. You don't just get the excitement of winning, you get the visceral feeling of finality when you tear up cards and mark the board. I understand that that experience is not for everyone, but there is a lot to recommend it.