I know that ostensively, the per adventure/AP Background choices are supposed to fill the role that campaign traits used to fill.
My issue with that is that to benefit from that, you have to reduce all your available backgrounds down to the ones provided for that AP, meaning all the core backgrounds and other books' backgrounds are competing with integrating your character into the story.
It is kind of back to the same problem when the equivalent of skill feats and combat feats were taking the same resource pool. Backgrounds are pulling from AP specific (giving you a tie in) competing against a background you read in a book you got that inspired you to make a particular character, but now have to decide if you complete that character with that background, or change to an AP specific on so you are better lined up with the story.
In the end, actually, from a wider rule standpoint, it occurred to me that we already have a system in published rules that can, and would fix this.
PFS uses Boons. If APs and Modules had a list of Slotted boons that each player may choose one. It can grant them some minor benefit in the particular adventure, and more importantly, it can define some sort of connection the player will have to the plot of the adventure.
They can use established boons for establishing acceptable power/scope for these boons. Limited use abilities that may grant a bonus in certain circumstances a certain number of times. Or a niche granting of a particular skill in certain circumstances.
For instance, perhaps a boon could involve growing up in a particular town, and knowing everyone in it. This allows the individual to be treated as they are trained in society when using the skill in relation to those in their hometown. (or if they are already trained they get a circumstance bonus in the same set conditions)
Another boon gives them downtime access to a skill check of a skill they don't know, by having a friend. That friend can be defined in the boon and can be their tie into the story.
Some boons might even have a 'hidden' (but logical) effect in the adventure allowing certain challenges to be automatically succeeded by a person with a certain boon. Someone with the Local Socialite boon might auto-succeed a skill challenge to get invited to a local party or auction party. Because of who they are, they automatically can succeed. Otherwise someone might have had to make a hard DC check of some sort as one of the methods to access that encounter, for instance. Likewise, being the local shop-keeps friend means you start out friendly with them and get certain information automatically when you stop by to visit and ask. Otherwise someone may need to mention the right thing, or get some appropriate Diplomacy check to get the needed information.
These Boons could be selected even by members of a party who just left a different AP, potentially having used that first AP's background. The APs specific starting boons could be listed out and each player select one. It gets slotted for the duration of the adventure and helps define the tie for them in that particular adventure. Their boon in the first AP may have defined that they grew up in that adventure's town. The second 3-part AP might have a Boon available that indicates you had a cousin who moved to this town, and you haven't heard from them in a surprisingly long time, so you are investigating.
They even become something you could give out as rewards for certain achievements during an AP or adventure. You just saved this caravan, and the merchants. You have just unlocked the 'Merchants of Venison Friend' boon that gives some effect for you if you choose to slot it.
Basically it is leveraging rules that already exist, just not ones being used by the core rules. Instead it is using mechanics tacked on to support the Society Play, but could easily be leveraged to help insure characters get hooked into the storyline, for starting characters, as well as higher level adventures where the characters may have been originally created for an earlier adventure.
I'd just like to point out that having a campaign specific background is a give and take thing with the GM.
You limit your background choices in exchange for having much better odds of your background actually mattering, and perhaps grabbing the spotlight more frequently because if it. The GM has a much easier time of catering to a small handful of backgrounds specifically written for the campaign, and will appreciate you meeting them halfway by taking one of those options.
If you decide you want to run one of the many other backgrounds, well, you trade away some of the potential background-linked-gameplay for having a more "your own" character. Your unique background might shake up the gameplay in interesting ways, as your choice clashes with what the adventure might expect, so at least there's that.
But overall, I think your concept is neat. It would provide mechanics that support the discussion that should already take place about why each person's character would be tied into the story.
I just want to point out that just because you don't choose a background from the AP's guide that will prevent the character from being "better lined up with the story." For example, is a character that selects Scavenger or Street Urchin as their background and Goka as their home city (trained in Goka Lore) really that much less "integrat[ed]... into the story" than one with Newcomer in Need? Or is Attention Addict (focused on fame) anything more than a slightly more specific take on Gladiator (which could involve other motivations if the player so desires)?
Backgrounds are a tool. The AP-specific backgrounds exist as examples and options to quickly tailor a character to that AP's themes and story, but a little tweaking of "generic" backgrounds' flavor text and/or adding (non-mechanical) specifics that align with the particular campaign can provide much of the same result.
One of the GM's jobs has always been helping integrate PCs into a given campaign. The GM knows much more about what motivates their particular players and the PCs involved; for every group that wants a relative as the hook, another wants a different type of hook.