That's a good idea.adding a diode and small capacitor to each stage would guarantee that the activated stage would stay set after the common reset signal had returned high.
It's also called a radio push-button circuit, and that modification should make it reliable (I didn't realize that was what the TS wanted).
Here's the LTspice simulation:
I added a resistor and a capacitor.
I selected the opposite latch output for the signal, since a high output for the selected PB is more commonly wanted.
Of course the circuit can be expanded indefinitely with added stages.

