How do you measure whether or not your platform engineering team is meeting their responsibilities?

I always like the burndown chart, but this, of course, requires proper sprint planning with actual estimation and scoping. Here is where many orgs fall down - they don't estimate but without (a) a prioritised and well-scoped list of items, (b) measured engineering capacity, and (c) estimations against items, your sprint plan will only be a guess or a best-effort. In those cases why do sprints at all if you are spilling tasks over every single time?

BUT if you do these things, the burndown chart will aid you every day in seeing progress, and in pinpointing - early - when things might be going awry.

Both through qualitative and quantitive analysis. Qualitative through interviews with relevant stakeholders about how the team is meeting their expectations. This helps identifying 'emotive' trends in perceived performance and helps identifying new expectations. I have encountered plenty of situations where the performance dashboards were all lit up green and the stakeholders were very unhappy. So that takes us to quantitative measurements: make sure that what you capture gets communicated. Different stakeholder groups have differing information needs - so give them only the information they need. Create different dashboards for different stakeholder groups. And keep the information to the barest useful minimum.