Galactic core has no short-living stars and supernovae
Not true, there actually are some very young massive stars very close to the galactic center (less than 1 pc), so star formation really can happen there, though probably very rarely.
Although most of them are old red main-sequence stars, the Galactic Center is also rich in massive stars. More than 100 OB and Wolf–Rayet stars have been identified there so far. They seem to have all been formed in a single star formation event a few million years ago. The existence of these relatively young (though evolved) stars was a surprise to experts, who expected the tidal forces from the central black hole to prevent their formation.