Really-quick-and-dirty calculation time!

Let's say 3 supernovas per century and each sterilizing 10 light years in radius.

That produces an average sterilization volume of about ten cubic light years per year. Total volume of the galactic thin disc is on the order of 2*10^13 cubic light years. That produces a half life of sterilization on the order of trillions of years, though you can bring it down to billions if you increase the supernova rate by a factor of a thousand or increase sterilization radius out to 100+ light years.

We can probably discount the galactic core for any purposes though - I've seen fun papers proposing evidence that it undergoes periodic starbursts every few tens of millions of years and the galactic supernova rate then briefly goes up to something like one per year with most of them in the core.