Our use case is to have an, Azure hosted, web back-end send notification messages to our existing mobile Android and iOS applications. I note from the Azure Notification Hub (ANH) documentation show that we can send messages down to both Android (Google Cloud Messaging and Firebase Cloud Messaging) and iOS.
I also note that all the tutorial samples include you placing ANH-specific code into the mobile clients.
Will ANH work assuming we have only platform-specific calls (e.g. Firebase for Android and Apple Push Notification Service for iOS) in our mobile apps - and no ANH-specific code in the apps?
It is up to you and depends on your scenario/architecture. ANH as a platform doesn't require specific client-side code.
Here's a typical high-level architecture (which does have client-side ANH-specific code):
If you can design your system in such a way that the back-end can register PNS handles by itself (i.e. step 2 on the picture will be different), then you don't need ANH-specific code on the client. In other words, if you can somehow register tokens from the back-end instead of doing it from the client, there's no reason why NH would restrict you from doing that.
If I understood your case correctly, you already have a number of app users that you want to push to using ANH. You'll need to find a way to register tokens of those devices with NH. You can literally do it by writing a simple console app and running it from your laptop as long as you have those tokens available. And then use the Azure back-end in a usual way to do actual pushes, since the push part doesn't care how the devices got registered.