To the OP... For about $100 you can have Best Buy install an aftermarket alarm that is neutered, except for the auto-lock capability. All of my vehicles have keyless entry, and 'factory alarm' - but none of them will automatically lock the door when you close the door an walk away. Getting a simple, cheap alarm that ties into the existing factory system will give you exactly that. It doesn't need a key fob or siren, and it uses the (un)lock pulse from the door locks, and power to the dome light to know when to arm/disarm. It gets installed, and you'll never know its there.
With mine, 30 seconds after the dome light goes out (without the accessory or ignition power being on) the alarm sends a pulse to lock the doors. Simple. There is a small 'valet' button that you press to put it into override (for when you have service done, or a neighbor borrows the vehicle). I tie-wrapped an extra key in a hidden spot on the outside of the vehicle, so if I ever get burned by accidentally leaving the keys in the car, I have a way in.
Also, anytime I see anyone attempt entry into my vehicles like that, the first thing I do is call the police to inquire if anyone else in the neighborhood has reported a vehicle burglary. If so, I offer them a copy of the video (usually a YT link). It helped nab a convicted felon (on probation no less) that tried to get into my locked truck, and then went on to the neighbor's unlocked Jeep. Stole an iFruit charger, sunglasses and a couple other trivial items. The punk left a used cigarette in the cup holder - which was the only reason the neighbor even knew it happened. He then came to me to ask me to check my videos. Sure enough, we captured him on video. The police were called, and to my surprise, the tested the ciggy for DNA. 8-months later they arrested him as he was stocking shelves at the local walmart.