Ticket to Ride house rule to help players with short destination tickets
The goal in TTR is not "complete the cards you're given", it's "get the most points". There are plenty of ways to do this with short routes (e.g. by focusing on building only length 4+, not going for longest route, and picking the spots where everyone wants to go).

If no one in your group wants to adjust their play style to match the cards they're dealt, then you obviously don't want to play the game as intended. A rule change might be an outcome, but this perceived unfairness might be part of the process of getting to know the game mechanics.


  I (...) thought a good idea might be to allow a player to draw a new ticket after completing a mission


This rule comes with a drawback. You would have to show your completed routes, which is something you would normally do at the end of the game. I would not recommend this as it would upset yet another part of the game.