Is it possible to set the default view in the iPhone music app to be “downloaded” only?
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?