Firebase service_key json file contains sensitive information so mentioned in his video that you should never commit it to git. Another way to add the service_key is with environment variables.
FIREBASE_SERVICE_ACCOUNT_KEY='JSON file content without line breaks'
And in firebaseAdmin.ts file, change this
const serviceKey = require("./service_key.json");
into
const serviceKey = JSON.parse(
process.env.FIREBASE_SERVICE_ACCOUNT_KEY as string
);
And when adding the environment variable in vercel don't put single quote.