Do not track dev productivity. That is a concern of the engineering manager, not the PM. If a product manager needs analytics, dig into what they are trying to accomplish. Most of the time, they are trying to understand delivery velocity in order to plan roadmaps and communications, and do not truly care about productivity of individual devs. This is why Scrum got popular - not because it is good for the devs, but because it is predictable for the PMs.
So you are on the right track of pulling info from your system. Continue to treat this as a consulting gig - work with them on metrics they can pull from your systems to give them the answers they really want. Depending on how easy they are to work with, and how open to change, suggest that they change their communication patterns, and do not promise timelines to stakeholders or customers. If they limit customer transparency to roadmaps and priorities, it is amazing how little they need to track from the devs. And in my experience, the increased autonomy that gives the dev team results in feaster delivery, anyway.