Firstly, why are you using hosts? Surely DNS is a better way to implement host resolution?
You could always set a copy in the Startup/shutdown scripts section in GP to copy a host file of your choice, or wrap the host copy as an msi and snap it to the servers OU as a software installation? Just a few ideas off the top of my head.
The trusted sites setting needs to be imported, so first, completely configure the settings as you'd like them on your pc, then fire up Group Policy MMC and go to the setting in my screenshot, and click on the import section...