Non Watch or Phone GPS

The only thing you will really find, small enough to run with, are little tags made for a child's bag/clothes or a pet tracker. Neither will be as accurate as a watch or phone and most likely are useless for actually running (they don't really do pathing/distance, just ping every few mins, so it's less about the path taken and more about where you are).

Otherwise there is zero real use case for this.

Even getting a basic GPS watch (like a Garmin Forerunner 45) and putting it in your pocket without the band seems annoying.

There was nike+ which was a chip that was in trainers and you could use it with Ipod nano which is pretty small.

Don't know if these still exist.

Or you could do the bit at the end of your thread - buy a watch and remove the main bit, then superglue a clip onto it.

If you just want something to record pace/distance/splits/time, you could use a Stryd footpod. You run with it on your shoe and can use it without a watch or phone...and then after the run it will sync it to an app. It’s expensive and there’s no gps, but it will track everything accurately.