Update: I installed GBoard. It does utilize Google's servers instead of Apple's, and I am able to use dictation without getting the persistent spinning network activity indicator. In fact, I don't even briefly get the indicator (I'm trying this on WiFi at the moment).
Unfortunately, GBoard seems to be implemented in a rather clunky fashion in iOS. I can switch to it easily enough, but there should be an option to simply make it the default, rather than still have the stock keyboard with a key for switching between keyboards. More importantly, it's the GBoard dictation that is really clunky. After switching to GBoard, if I tap the microphone key, the screen for dictation pops up, but the service immediately stops with an error message of "Sorry, I didn't quite get that." If I return to the keyboard and try again, it happens again. The only way to actually dictate anything is to allow the error message to happen, and then tap the microphone button from the error message screen. Then I am able to actually dictate a message. To make matters worse though, I cannot see any of my dictation until I am done, and then the text populates all at once in the text field.
With Apple's stock keyboad, I can see the text populating as I dictate, which is exactly the way it should work, and is exactly the way GBoard works on Android. Third-party keyboards work properly on Android, at least all of them that I ever tried. I'm not flaming Apple here, I'm just wondering if GBoard's clunkiness on iOS is specific to GBoard, or if this is the way that Apple handles all third-party keyboards.
I think this definitively shows the problem is with Apple's dictation service. Is anyone else concerned that the persistent spinning icon is an indication that the dictation service never stopped, and the phone's microphone is still "on" and continuing to send data to Apple's servers??
Next step is to try and monitor cellular data use while the network activity indicator continues to spin.