What if one player claims a route between two cities before the player who’s got the destination ticket?
No, you must complete a route yourself, with your train pieces.
From the rules:
Points are lost if you do not successfully complete the route given on the Destination Ticket(s) you kept. [emphasis added]
Blocking off a city that a player needs to get to is part of the strategy, if you can figure out where they want to go.