The 12volt battery in the car powers the door locks and the trunk release. The rechargeable battery in the remote powers the remote.

The CA remote only has a single battery which is user replaceable. Sure it has the buttons on the remote which is done for several reasons:
1) You might want to unlock the doors from a distance; farther than what CA allows when touching the handle.
2) You might want to lock the doors from a distance; farther than what CA allows and also means you don't have to touch the door handle.
3) You might want to open the tailgate which means you might need to unlock the doors first.
4) You can close windows with the remote; this is with the CA or non-CA remote.

The CA remote and non-CA remote look the same with the exception of the battery door on the CA remote. If the remotes were the same, then why does BMW have two different part numbers; one for each? If the CA and non-CA remote were the same, then BMW could just sell a single remote and if the car can use the CA portion, then it would, if not then it would work as a normal remote. The internals between the keys are the same and BMW just redesigned the back panel. A different plastic back panel is not expensive at all to make. The internals cost more to make than the plastic back.

If a car doesn't come with CA then getting a CA key doesn't add the functionality. There are door sensors and antennas required for it to work.

Many SA's still say the CA remotes have a rechargeable battery. I have a remote that has never been in the ignition slot until well yesterday. It was the key that I used to drop the vehicle off at the dealer for "the campaign" and while I have replaced the battery in it; I replaced the battery in the remote that I use s I replaced both. Either remote would allow CA as well as locking and unlocking the doors via the buttons. So, if there is a rechargeable battery in it and BMW is saying that it takes 30 hours to charge and to use it twice a year; how come it has never been "used" but yet still works? So, what would the rechargeable battery do when the replaceable battery is running everything? In a non-CA remote, it is constantly getting recharged as it needs to be put in the dash. With a CA remote, when the user replaceable battery is dead, the CA functionality doesn't work and you cannot unlock the doors either. So, once again, what would a rechargeable battery do in a CA remote; it doesn't run the buttons or the CA portion, the replaceable battery does.

Now to your question. Since CA is a two step process:
1) Proximity: the remote needs to be near the car for it to see it
2) It requires the door handle to sense a hand or finger for it to lock or unlock.

So, when you walk by you have condition 1 met; proximity. Now you are not touching the handle, so that tells me you have a CA door handle that is bad. Another possibility, the CA control module is bad and isn't requiring condition 2 to be met. To test this, you could disconnect one door handle at a time and walk by and see if the problem goes away. if it does, now you know which CA door handle is the problem.