You job is to reach the destination 1.as soon as possible 2.before anyone else's does. Any one will make it meaningful. I think 'does' is understood so it is OK if it is not written.
All you need to do is reach the finish line before anyone else. OK
All you need to do is reach the finish line before anyone else reach it. OK, but it is 'reach
Your job is to reach the destination as soon as possible before anyone else does. Does is optional. You could put a comma after possible. With a comma, the sentence means you should be there as soon as possible and you should be there before anyone else. Without the comma, it means you