However, when I try to do Find My Phone, it only works if I have the app open.
That's exactly how it is designed to work. See also the document Using the Find My Phone Feature on a Garmin Watch | Garmin Customer Support
iPhone Users
If the Garmin Connect app has been fully closed, the Find My Phone feature will not work, as it requires that the Garmin Connect app be open or running in the background.
When I got my iPhone "Find my Phone" would only work for a while. I changed some settings probably both in the Settings app and in the Garmin app. After a while I got it to work. I cannot any longer remember which setting(s) made a difference. Maybe it had something related to iOS not closing apps...
Garmin Fenix 3, iPhone 13 Pro Max, iOS 16.0.2