When it comes to KPIs, revenue is usually number one. But how much weight should customer satisfaction carry? How do you measure this at your company?

IMO customer satisfaction should be a metric that is measured carefully and reviewed by senior leadership along with their other KPIs. Satisfied customers drive retention, referrals and repeat business. Customer churn makes growth difficult.  I once worked with a client that had a growth goal of 20% per year with a churn of 15% per year. This meant they had to grow at 35% to achieve their numbers. The new VP Sales knew he needed to stop the churn to be successful.

Landing a client is vanity. Retaining a client is sanity. Expanding a client is king.

Depends on how you measure satisfaction. I think satisfaction can be a fluff metric and just trying to get a point in time, emotional response. What makes more sense is measuring leading and lagging indicators for the impact that your product and service are having on the company, team, and people that are using it. Impact needs to be recurring in order to drive recurring revenue which ends up driving customer satisfaction. 

Customer Satisfaction is a leading indicator of future revenue. So you need to measure and action customer satisfaction / loyalty and then also measure revenue...

In the beginning, Customer Success is a cost center but soon it becomes a profit center. 
It is important to measure the right KPIs to have an overall understanding.
Start with the churn rate and also look at the NPS and the feedback the Customer Success Managers get. 
But you should also talk to the team to understand if anything is missing such as technical documentation or support for another department. In most cases, there are multiple reasons for a high churn rate and they are hard to identify by just looking at one KPI.