If you don't trust your hardware, it's almost always game over. Desktops have devices running dubious firwmare as well, but at least with a hardware key store, the window of compromise ends at the time you update – a stolen key stays stolen forever.
That's not to say that you should trust modern smartphones. That's up to you. It's just that in whatever "trust" means to you, the baseband urban legend shouldn't come into the equation.
If there are vulnerabilities on the AP (main processor) then you can hack it from the baseband, but also from the bluetooth or WiFi chips.
Caveat emptor on cheaper/older phones or ones you enabled developer modes on.
Yes, either the Browser or the OS will need to be involved. For example, for Webauthn on Chrome on Windows: chrome receives the request, then it calls the Windows Hello APIs. Then, that Windows Hello API shows a popup to read a physical security key or authenticate a virtual security key via face/PIN (this is protected by a TPM, but it's "virtual" since Windows generates it via the TPM but stores it encrypted on-disk).
To support a syncing fido keyvault, Chrome could very well redirect the calls to its own popup for choosing to 'use Chrome', or 'use another key', which would then call the Windows Hello API. In fact, Chrome already supports this[0], with 'Add a new Android phone' is simply how they're presenting Webauthn over BLE, and it works with iOS when passkeys are enabled in the iOS developer menu[1].
0: https://i.judge.sh/Qq93C/v_GG5R7LyM.png
1: https://developer.apple.com/documentation/authenticationserv...
(I don't even want to think about how to tell Mobile Safari on my iPhone how to find my key)
EDIT: My ideal setup, I think, is an app on my phone that I can use as my token - somehow signaling to my desktop/laptop that it's nearby and can be used as a token and ideally popping up a notice on the phone lock screen when there's an authentication request so I can quickly get to it. Then in my app, I'm free to export and backup my keys for all of the sites I'm enrolled with as I see fit. I know, I know, maybe being able to export the keys makes the setup less secure, but I will trust myself not to accidentally give the backup to a phishing site. (And I do worry that I'll accidentally get phished using a TOTP app, so I'd like to switch to FIDO, but I don't want the pain of multiple keys)