How about starting with mean time to merge a code change? I get that there are other variables that contribute to productivity but things like satisfaction, collaboration etc are extremely difficult to measure well and just IMHO pretty tangential (disclaimer - I work on a dev prod team and my entirety of last year was spent building engineering metric dashboards and discussing what to measure, and it's not easy so I…
I think that qualitative metrics like satisfaction and collaboration could be helpful, especially when combined with traditional metrics like mean time to merge a code change. Taking my previous example of overworking or cutting corners to achieve high numbers, a qualitative metric for something like satisfaction might indicate a problem where a work output metric wouldn't.
But I think that any combination of metrics will be an oversimplification that could lead to problems if they are the only thing that matters. I'm not sure where the balance lies. I like that metrics can offer an objective view of performance and make it easy to spot trends. But I am wary of them oversimplifying things and dehumanising the team.