I've opened a DTS ticket to get some clarification on this but while I wait to hear from them, I've done some additional investigation and have observed something interesting. Xcode has a watchOS Simulator titled "Apple Watch Ultra (49 mm)". When I deploy the code to this simulator, it behaves as I would expect it to. That is to say, after granting "always" authorization in the second prompt, I get one more callback to locationManagerDidChangeAuthorization(_:) which indicates manager.authorizationStatus == .authorizedAlways and then the authorization prompts cease. I am now able to start location updates and receive them with the app in both the foreground as well as the background. Opening the Settings app on the watch simulator and navigating to Privacy & Security → Location Services, I see the app listed and it indicates that it has "Always" access to location data. Tapping that row shows that the settings ("Never", "Ask or When I Share", "While Using the App" and "Always") are still disabled but again, "Always" is selected and the app appears to function properly.
There is another simulator in Xcode (listed under iOS Simulators) titled "Apple Watch Ultra (49 mm) via iPhone 14 Pro Max". If I deploy the code to that simulator, I experience the behavior I described above in my original post. This is the same behavior I encounter if I deploy the code to a real Apple Watch Ultra that's paired with my iPhone. So now I'm wondering if, despite being an independent watchOS app, this behavior is somehow related to the watch trying to use location data from the paired iPhone? As I said above, the location authorization prompts only appear on the watch (I've tried keeping the iPhone nearby and unlocked when running the watch app but still don't see any location authorization prompts on the phone). Could the prompts on the watch (after the first series of prompts, I mean), actually be asking for location authorization on the paired iPhone? If that's the case, I guess I'm still not sure why my selection(s) in the authorization prompts aren't being applied to the phone but at least it would give me something to go on.