Haha I was reading the entire thread because Im pretty sure I have your solution until I got to the bit where you said the car is written off...Now I guess we will never know...
Between myself and my friends there are many W124's and they all have an option where you can set the central locking somehow to only unlock only the drivers door when unlocked using the key. When the engine is running the central locking operates all doors at the same time. This is a 'safety feature' that can be disabled and you probably enabled it by accident. Your description matches the functioning of this feature exactly.
I installed a remote system on my central locking so have never used this feature. It should be in the owners manual, if you dont have one look online.
Maybe someone else with a manual or knowledge of this feature can tell us how...