How does the ESPHome Dashboard determine if a node is online?

Add the environment variable ESPHOME_DASHBOARD_USE_PING=true as @finity noted above. Also documented here Frequently Asked Questions — ESPHome.

To add the variable with HA addons: ssh into your HA then open the addons.json file.
Look for "description": "ESPHome add-on for intelligently managing all your ESP8266/ESP32 devices", then append:

      "environment": {
        "ESPHOME_DASHBOARD_USE_PING": "true"
      },

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Documentation on Add-on config: Add-On Configuration | Home Assistant Developer Docs