My issue is that I don't want to proceed to do this if it affects other teams who may be on related profiles. Will other teams' developers profiles stop working because my device gets added to the profiles? Or is it always completely safe to tell Xcode to "fix it" when it reports that (my new test) device is not listed on any of the valid provisioning profiles?
It will not affect other developers on the team or invalidate their provisioning profiles. It will simply generate a new profile with your new device added. Next time their Xcode syncs with apple's servers, they will get the updated profile with your device listed as an approved device.
Things that can affect the other developers:
Adding a new device when your account is close to the 100 device limit. If your account has 99 devices and you add your device, your team will be unable to add any more devices until you renew your account and remove old devices to drop back down under 100.
Invalidating the current certificate and creating a new one, then regenerating the provisioning profile. If you do this, it will break any old builds using the old certificate. I don't think Xcode will ever do this for you, though; you would need to do this manually through the apple developer site.