Last I checked the campaign traits were not 'required'. The way we play with our current pathfinder group, one of your starting traits must be a campaign trait. After this point if your character dies and you bring in a new one, you select your own two traits normally.
it’s a way to quantify (and encourage) building a character background that fits into your campaign world. Think of character traits as “story seeds” for your background; after you pick your two traits, you’ll have a point of inspiration from which to build your character’s personality and history. Alternatively, if you’ve already got a background in your head or written down for your character, you can view picking his traits as a way to quantify that background, just as picking race and class and ability scores quantifies his other strengths and weaknesses.
To answer your question, there is no need. Ultimately it comes down to you as the DM to decide. Just keep in mind that the purpose of the traits is to promote and encourage development of a characters background.
For the people in the comments that were requesting the source, I linked it in the comments below but incase if you did not c atch that you can find the blockquote above on the pathfinder srd: d20pfsrd.com/traits.
For those who also had questions as to why I require my players (As a DM) to select a campaign trait for adventure paths and not when they re-make characters: simply put, it just makes sense. If you have ever ran an adventure path and one of your players died and had to remake a character, then you will be able to relate to the OP's question.
For adventure paths like Jade Regent, or Carrion Crown for example, it does not make sense for a starting character to be randomly invited to a dead mans funeral, and being mentioned in his will, without any history or relations to him. The campaign traits for Carrion Crown help introduce the players into the story so that they start the adventure with a purpose and goal. The same is true with the other adventure paths, Jade Regent for example you can choose to be a sibling of one of the NPC's. When you die however, does it make sense to bring in a character on the other side of the world that "Knew the professor, and was also there for his Will" (dead man), or "Surprise I'm a sibling of you too!". It doesn't, this is why the campaign traits are recommended for starting characters.
I would also like to apologize to HeyICanChan for when they first asked why I was giving replacement characters more freedom. I did not fully understand why this was being asked. It sounded to me like a question that a Player would have for their DM, "Why are you disallowing us to play X Race? or X class variant?". My answer would have been situational based off of the campaign that the DM was choosing to run. Perhaps that specific race or class does not fit into the storyline? Just as selecting a non-campaign trait does not give a character any incentive or or reason to be included in many of the early game scenarios.
The rules are more like guidelines, if you are a player and feel that none of the campaign traits fit your characters background, then talk to your DM about it. Campaign traits do not need to be restrictive.