Was GRAVITY built to look at one star?
This link gives you all the papers that have cited the instrument description paper.
The list of papers shows that it has been used for studying: the centres of AGN, close binary systems, discs around young stars, the atmospheres of AGB stars and interferometric imaging of exoplanets at least.
Here is a paragraph from the instrument description paper

Inspired by the potential of phase-referenced interferometry to zoom in on the black hole in the Galactic center and to probe its physics down to the event horizon (Paumard et al. 2008), we proposed in 2005 a new instrument named GRAVITY as one of the second generation VLTI instruments (Eisenhauer et al. 2008). At its target accuracy and sensitivity, GRAVITY will also map with spectro-differential astrometry the broad line regions of active galactic nuclei (AGN), image circumstellar disks in young stellar objects and see their jets evolve in real time, and detect and characterize exo-planets especially around low mass stars and binaries – in short we will “Observe the Universe in motion” (Eisenhauer et al. 2011).