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.

I would characterize it as the things the government would lie about are different than the things a company would lie about.