What are the drawbacks of using a GFCI instead of a proper ground wire?
Correct.  

GFCI is actually the ideal solution for the people problem (shocking people) and serves that purpose better than grounding.  Assuming no boneheaded bootlegging of grounds.  But GFCI (alone) does nothing for the ESD/lightning problem which can damage sensitive electronics or cause goofy problems with radio transmitters.  

Current travels in loops.  Current wants to return to source.  For natural electricity like ESD and lightning, the source is ground, not neutral.  

You need fair warning of the lack of equipment ground, hence the labels.  The labels also allow you to pass inspection on 3-prong outlets without a ground.