Compromise between privacy/security and losing a phone

I fail to see it helping if stolen. It’s not enough evidence for a warrant since it can only tell the general area and you are not planning to go recover the phone at night batman style. Phones also have the unique imei already as well as the serial number.

If lost, one option would be to use an external tracker attached to it, or even a remote controlled beeper. If you search online for “remote finder beeper” you’ll see a lot of things. They are more for keys and usually attach to a key ring, but with some effort you might be able to attach one to a case. There are also phone cases that come with key rings.

The beepers work like a garage door opener, except starts making a loud beeping sound when you press the remote while in range. This is good if you have a tendency to know the general area where you left things but not the exact location. This avoids the issues with having a GPS tracker.

Another thing you can do is have your contact info somewhere on the case itself. There’s a decent chance that whoever finds it will call you or send a text message letting you know. Most people aren’t thieves.