The public sector doesn't have a monopoly on internal politics. Posturing between departments to be included in the next big initiative happens in lots of companies.
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.
