Why aren't more contact tracing apps using GPS instead of Bluetooth?

Many contact tracing apps are only trying to use Bluetooth to decide which devices you're near.

Is that really a good approach?

I don't think so. It doesn't really matter if you're physically near me, if I cough on something and an hour later you touch that, you're very likely to get infected.

Shouldn't the apps trace a persons movement? Then once a case is encountered ping all the individuals who crossed paths?

The persons movements wouldn't necessarily have to be sent to a central server, they could be stored locally.

Are there other concerns or limitations that I'm missing?