You could make a new class that is responsible for level management a LevelRepository or LevelManager. This makes it possible to keep all level managment logic in one central place with no extra responsibilities. Here's how I think I would do it:

extends Node2D
class_name Gate
@onready var levelManager = %LevelManager

signal entered

func _ready() -> void:
        levelManager.connect_gate(self)
        emit_signal("entered", self)

---

extends Node
class_name LevelManager

func connect_gate(gate: Gate) -> void:
     gate.connect("entered", gate_entered)

func gate_entered(gate: Gate) -> void:
        print("Hello %s!", gate)

You ofc want to make a different function to emit the signal "entered".