The goal is to have users with admin privileges
Since you are using the Authentication service you already have half of the solution: with authentication you can identify each user who is using your app.
The other part is Authorization: this is normally done with Security Rules in Firebase, both for Firestore and Cloud Storage.
To be able to authorize certain users (identified through authentication) with Admin privileges, you need to know which users have the admin role in such a way you authorized them to execute the admin functions.
One possible way to identify the admin users is to have an isAdmin flag in some user documents in Firestore, as you mention in your question. There is an example of Firestore Security Rule using this approach in the documentation.
HOWEVER, you will encounter some problem if you want to use this flag (stored in Firestore) with Security Rules for Cloud Storage. At the time of writing, it is not possible to read the value of a Firestore document in Security Rules for Cloud Storage.
The solution is to use Custom Claims. You will find all the details in the doc on how to implement it in such a way it fulfill your needs.
Can I build an Admin Panel inside the client app?
Yes, you can very well do that. As soon as your security is correctly implemented (through Authentication and Security Rules, as explained above), there is nothing that prevents you to develop an Admin panel. If a user that is not admin can access the Admin panel, he/she will not be able to perform the admin actions (i.e. writing/editing/removing Firestore or Cloud Storage data).
Moreover, with Custom Claims, you can access them in the front-end to modify the client UI based on the user's role or access level (i.e. showing the pages, buttons and menu items of the Admin module only to admin users -note however that this does not prevent someone to reverse engineer your app and execute the queries dedicated to admin users: this is why it is key to correctly implement the Authentication and Security Rules parts-). See this section in the Custom Claims doc.
Should I build an Admin Panel in another app and using Cloud
Functions?
If you don't want to over-complexify your app with some logic to hide/show the Admin panel elements (based on Custom Claims, see above) you can very well build the Admin Panel in another app.
If you have specific needs/access restrictions that cannot be implemented through standard Security Rules you could very well use some Cloud Functions to check the user is an admin and to execute the writing/editing/removing admin actions (note however that while it is quite easy to interact with Firestore from a Cloud Function, it can be a bit more tricky with Storage: using the Cloud Storage Client SDKs is much easier than interacting with Cloud Storage through Cloud Functions).
You would preferably use Callable Cloud Functions, since "with callables, Firebase Authentication and FCM tokens, when available, are automatically included in requests". (See https://firebase.google.com/docs/functions/callable).
Side Note: You may be interested by this article, which details how to to create an Admin module for managing users access and roles. (Disclaimer: I'm the author).
