Hi and welcome to the forum.
In that you live in the Bay Area, you have several world class hospitals to choose from, as well as several top notch surgeons. I'm in Southern California, and was in a similar situation, with LA only 90 minutes away and San Diego 60 minutes drive, I had several choices. We are fortunate to have so many choices, as some don't have any top hospitals nearby.
I'll share with you how I narrowed things down. US News and World Report ranks the top hospitals every year, based on specific medical areas. I started there looking at their rankings of Best Hospitals for Cardiology, Heart & Vascular Surgery. See link below.
https://health.usnews.com/best-hospitals/rankings/cardiology-and-heart-surgery
There were several top hospitals on the list nearby me. I then looked up specific surgeons at these hospitals, to see who was the most experienced and had the best outcomes, meaning the lowest mortality rates. This data is reported and there are resources which can provide it. I can't remember offhand the name of the website I used, but if I think of it I will circle back and link it. It was free, but at some point they required you to join to see the stats.
The top hospitals will have generally excellent surgeons, but there were some differences in terms of experience and outcomes. I narrowed it down to the head of heart surgery at Cedar Sinai and the head of heart surgery at UCLA. Both had over 3 decades of experience in heart valve surgery and both had survival statistics much better than the national average, as well as hospital re-admittance rates lower than the national average. Keep in mind that the survival stats are a 30-day survival rate, and so having a top score here speaks to the surgeon's ability, but also to his team, including the ICU care team. Like many people who go through valve surgery, I had a complication and was very glad that I was in competent hands in ICU, to see me through those rough waters.
I set up a consultation with the surgeon at Cedar Sinai first. He didn't give me much time, and mostly just gave me a small lecture about what aortic stenosis was and my valve condition. Later, I would meet with the UCLA surgeon, who gave me over an hour of his time and answered every question that I had. He was also happy to discuss published medical literature about my condition and demonstrated a depth of knowledge, giving insight into studies which I had questions about. I felt very comfortable with him and decided he was the one. I had these consults months before I was told that it was time for surgery. When an echo months later indicated that it was time for surgery, after meeting with my cardiologist, I scheduled another consultation with the UCLA surgeon to prepare for surgery, discussing the final choices. You will want to specifically ask about how experienced they are with your situation- mitral valve repair and replacement. You also may find that the top hospitals have a particular surgeon who is the "go to" person for repair, when that is a possibility.
I don't think that either of the surgeons saw the first consult this way, but I looked at them as me interviewing them for the job of opening up my chest. I think either one, along with their teams, would have been excellent, but I went with the one who seemed to care the most and was willing to take as much time needed to answer every question that I had.
One other point. Don't take too long deciding on which surgeon. You have indicated in a previous post that you have symptoms. If you have your eyes on a particular surgeon and he is booked for the next 6 months, move on. There are probably over a dozen high caliber surgeons in the Bay Area and probably the biggest mistake that you can make is to wait too long.
Best of luck in choosing your hospital and surgeon. Please keep us posted.