The idea is a good one, but there are many flaws in it.
One, you would have to trust the person (human) to attach the bungee cord to the car seat & keys every time they drive the vehicle (assuming the cord is not a "permanent fixture" to either item), which in a busy schedule, will easily be one more thing forgotten to do in the rush to save time or found to be too much of a hassle, so simply ignored (who hasn't purchased a item to make life easier to simply find it's too much of a hassle and never used it?).
Two, a FF child can easily get their feet tangled in the bungee, get their feet under a nonflexable plastic tubing (kids are amazing and can seem to undo or get into anything....mine chewed a snap off the adjustment strap of her carseat that was tucked away between the seat and the carseat when she was RF, have no clue how she got it out, but she did, chewed on it and ate the snap....) that is attached at the base or bottom of the car seat, causing undue stress on the ignition lock cylinder/switch (this is why people with heavy key chains have more instances of failed lock cylinders/switches due to the stress on them, they were not designed to carry a lot of weight), which can and will increase the instances of a failed component, which may and can leave the family stranded.
Three, even with a long enough cord, it can easily become something even a parent could possibly get tangled up in when reaching back to grab that dropped sippy cup or hand the child something, causing more distraction to the driver to get untangled. If routed along or below the seats, that would add even more complications to hook it up to the seat and keys every single time the vehicle is used with the child in it.
Four, if the bungee becomes under stress (from like a kid playing with it with their feet, those kids that are limber getting it to their hands), the bungee snaps, that hurts.....BTDT, they fail and become weak overtime, and again would have to be check frequently to make sure they don't become weak....another thing to be forgotten.
I totally commend you and your wife on trying to find a solution to this that is happening more often.....unfortunately the solution is not going to be a easy one or even if one is found, will still not be fail proof.