I don't understand how my google action shows up on "Works with Google" search - google-cloud-platform

Question

I have followed below tutorial to build my own home action.
https://github.com/actions-on-google/smart-home-nodejs
I don't understand well on how this works. Let me explain you all about what I have understood so far and what I have done by now.
What I have DONE :
I built a conversational action (built with dialogflow's inline editor and intents) to change a data of the Firebase realtime database. (for instance, when I say "Turn the light on", then the Firebase data 'LED/OnOff' turns 0 to 1). Below is the code I have in my Dialogflow inline editor. (I have excluded the field including my project's credentials.)
'use strict';
const functions = require('firebase-functions');
const admin = require('firebase-admin');
const {WebhookClient} = require('dialogflow-fulfillment');
const {Card, Suggestion} = require('dialogflow-fulfillment');
admin.initializeApp();
process.env.DEBUG = 'dialogflow:debug'; // enables lib debugging statements
exports.dialogflowFirebaseFulfillment = functions.https.onRequest((request, response) => {
const agent = new WebhookClient({ request, response });
console.log('Dialogflow Request headers: ' + JSON.stringify(request.headers));
console.log('Dialogflow Request body: ' + JSON.stringify(request.body));
function welcome(agent) {
agent.add(`Welcome to my agent!`);
}
function fallback(agent) {
agent.add(`I didn't understand`);
agent.add(`I'm sorry, can you try again?`);
}
function handle_heatOn(agent) {
const state = agent.parameters.heat_on;
console.log(state);
var heat = 0;
if (state == 1)
heat = 1;
return admin.database().ref('0/OnOff').update({
on : true
});
}
function handle_heatOff(agent) {
const state = agent.parameters.heat_on;
var heat = 0;
if (state == 0)
heat = 0;
return admin.database().ref('0/OnOff').update({
on : false
});
}
//some more functions, related to controlling 4 different devices (heater, cooler, exhaust fan, LED) - exempted
// Run the proper function handler based on the matched Dialogflow intent name
let intentMap = new Map();
intentMap.set('Default Welcome Intent', welcome);
intentMap.set('Default Fallback Intent', fallback);
intentMap.set('heat_on', handle_heatOn);
intentMap.set('heat_off', handle_heatOff);
intentMap.set('cool_on', handle_coolOn);
intentMap.set('cool_off', handle_coolOff);
intentMap.set('exha_on', handle_exhaOn);
intentMap.set('exha_off', handle_exhaOff);
intentMap.set('led_on', handle_ledOn);
intentMap.set('led_off', handle_ledOff);
// intentMap.set('your intent name here', yourFunctionHandler);
// intentMap.set('your intent name here', googleAssistantHandler);
agent.handleRequest(intentMap);
});
Then I have deployed my action and Google has approved my action! It is now in production and can be searched in anyone's google account.
I have set up an OAuth 2.0 server. I'm really a beginner in developing, so I managed to use Auth0 to make the server. I have followed the below tutorial :
https://v3.jovo.tech/tutorials/google-action-account-linking-auth0
As a conclusion, when I invoke my conversational action in anyone's Google Assistant App, It will prompt the user to the authentication (login) screen (By Auth0). After the user inputs ID and Password, google assistant will link the user's account and launch my app.
However, my action is not shown on Google Home's "Works with Google" category. After trying to complete Brand Verification in Google Actions Console and linking one of my existent app on Google Play, I still don't know what to do or where to start from.
What I have understood so far :
Sorry for my shallow understanding in advance. I have understood that once my action is invoked by a user, it has to do account linking, which requires OAuth 2.0 server to exchange authentication token. If exchanging token has succeeded, the user's account is linked and he/she is able to use my action. The action will be launched after that.
I have searched the Internet and found some information such as Deep Linking and App Discovery, but I'm not sure this is the right keyword to start from.
I know I am really lacking in knowledge here. Please kindly at least hint where I should study and start.
What I want to understand and do :
I want to make my action, which is in production, show up on Google Home and Google Assistant App's "Works With Google" category without the prefix [test]. In other words, I want to let my app be searched just as the other company's apps. I understand that if you click on one of the apps of a company, your account will be linked and the company's action would be launched. Maybe I have to add some lines on Dialogflow's inline editor?? I know I might be silly, but I really appreciate your help. Thank you for reading my post.