I wouldn’t consider this dire- you would be surprised how often this happens. It’s usually not a deal-breaker, especially if the rest of your application is strong and personalized for the (right!) school, which it sounds like it is.
It’s good to remember that the AdCom is comprised of real people that understand that mistakes/typos happen so try not to beat yourself up over it (easier said than done, I realize).
I would definitely recommend that you do NOT reach out to the school or send an update about this. You don’t want to draw more attention to it, especially since they’ll be reading your application pretty quickly and it’s possible that they may not even notice.
Hang in there and know that there are a lot of moving parts throughout the process and rarely does the final decision come down to any one thing.
Posted from my mobile device