Interesting. Do you think private exploration solves this problems?
I think the main problem of the current setup is that the science teams are not balanced enough.
The radar sounding science team is mostly geophysicists because the interpretations are geophysical in nature, but there are not enough people who are experts in radar sounding, radar system and radio frequency wave propagation in general. The reason for that is, that this is not considered science and is looked down upon as mere "engineering".
Privatized exploration would initially solve some of these problems for sure, but once a group of people and its structure has manifested I believe they would eventually suffer from the same problem.
Anyway, this is a pretty complex topic which covers many aspects such as research funding, incentives in academia, vain egos, etc.
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.
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.