TL;DR

No. We can't toggle user consent locally without Apple. i.e., the consent status now for Analytics has a single source of truth (Source).

Tracking consent without Apple's Framework

If I have not received permission from a user via the tracking permission prompt, can I use an identifier other than the IDFA (for example, a hashed email address or hashed phone number) to track that user?

No. You will need to receive the user’s permission through the AppTrackingTransparency framework to track that user.

Tracking consent with a Website instead

If a user provides permission for tracking via a separate process on our website, but declines permission in the app tracking transparency prompt, can I track that user across apps and websites owned by other companies?

Developers must get permission via the app tracking transparency prompt for data that’s collected in the app and used for tracking. Data collected separately, outside of the app and not related to the app, is not in scope.

Alternative

The iOS Settings -> Privacy -> Tracking menu from within an app is not accessible because URL schemes to Settings were deprecated in iOS 10+.

But you don't need to go that far:

if let appSettings = URL(string: UIApplication.openSettingsURLString) {
    UIApplication.shared.open(appSettings, options: [:], completionHandler: nil)
}

The app page in Settings also has the option to toggle consent just for your app. Remember to track your consent status with Apple's consent for every app launch since the user can change this value at any time. For more info, see here.