Let's have the government collect the data and private companies can engage in that as well if they wish - both parties can call the other out if there's a discrepancy.

IMO the government's incentives are generally better aligned with truth telling but there are reasons[1] that independent studies may still catch the government out.

1. Famously, up here in Canada, Stephen Harper suppressed accurate dissemination of climate data during his administration that was only really discovered through independent analysis.

They do. Often for different kinds of things. It’s not “government good, private bad” or the other way around. Both are facile views.