Is it possible to set the default view in the iPhone music app to be “downloaded” only? On iOS 11, the music app has created a sub section for music actually stored on the device rather than in the cloud.

I never want to use cloud tracks, so I would like this to be the default view, but on my phone it always jumps back out to the top level menu. On my iPad, the menu setting seems to stick, though.

Is there a way to make it stick for the phone as well?
I'm guessing you must be quitting the Music app when you're not using it (or perhaps rebooting your iPhone or something similar).

Assuming you have the Music app on the Downloaded Music screen, or are playing music from within that section, then Apple Music should stay where it was last at. However, if you're in the habit of quitting the app or rebooting your iPhone, then it'll default back to the initial screen.

If the above fits your scenario, then there's really no need to quit the app. You can just leave it on in the background, even when you're not using it.

For more information you may want to refer to this question: Why is it better for an iPhone’s battery to NOT close down apps?