1) If you look at a GFCI, there is no way that the GFCI could open the ground.
2) It is permissible to install a GFCI on an ungrounded circuit provided it is marked as such "No protective ground".
3) Some testers will get confused. My expensive >$200 Ideal SureTest said I had reverse polarity, when, in fact, the ground was open, but it also indicated a high ground resistance of > 16 ohms or so.
It got the ground right, but the reversed polarity wrong. The ground was likely picked up through the shield of an antenna coax to the ground block some 40 feet away.