That's not guaranteed. Really it's a question of incentive structure more than public vs private.

For example there are cases I can point to of poor competition and high costs in NASA picking private companies to do big cost-plus development contracts, and cases where they did fixed-price contract bidding among a larger pool of competitors and got much better results.

While that might be true the costs of going up there are so high that there won't be 10 companies who get a crack at it.

Besides, the radar isn't the only instrument on the rover. There aren't many companies/groups in the world who have the know-how and financial means to do what NASA does here.

Or! They all produce garbage as they chip away at costs, and make bank telling people what they want to hear. The incentives are all wrong.