How do you assess the way an individual software staff member’s performance impacts their team’s success, especially in regards to team culture?

Look at the delivery for the team with and without the engineer. Look into the specifics of how he/she has contributed. Someone might be very good as an individual, but might be detrimental to a team, while some other cannot be effective on their own. Both needs to be addressed. 

I'm going to pull from Verus Global curriculum again, as they're a partner we leverage for cultural training/transformation tools. They have a concept called the Best Ever Principle. I think it would be helpful for this.

Here's a video that reviews it: https://www.youtube.com/watch?v=UilUI_Fw8gc

Basically it's about showing up as the best version of yourself whenever possible. I think the way you facilitate this is through a team workshop. Most people don't respond well when approached individually about how they're showing up (at least not initially). You could organize a team workshop that talks through the toughest scenarios and asks "how can we show up best in this moment?"