It's literally in the first paragraph of the link I provided:
Godot's scene system, while powerful and flexible, has a drawback: there is no method for storing information (e.g. a player's score or inventory) that is needed by more than one scene.
Using Singletons, you can create objects that:
- Are always loaded, no matter which scene is currently running.
- Can store global variables such as player information.
- Can handle switching scenes and between-scene transitions.
You don't use singleton to reload a scene, you use it to store persistent data.
OP already knows how to reload a scene, and was asking how would he reload his game while maintaining some portion of his nodes/data between those reloads.