One thing that is never mentioned when it comes to "price tag" articles for NASA projects, is how good of an investment it is. Most of NASA's research projects (especially the ones of the JPL) are done in America by American workers and companies and, therefore, boost the economy. Investments like this also strengthen the scientific community which has a ripple effect on education and future innovations. It is also absolutely necessary if the US wants to continue being the leader in the aerospace industry.
Outsourcing r&d work to private companies means, that innovations will be kept proprietary and patented behind growing price tags that will always be just a little cheaper than doing it from scratch. It also means that shareholders of those contractors get significant margins that won't be injected back into the system. In that sense, privatization doesn't reduce the "price tag" on research projects, it just makes some of the costs less visible for shortsighted financial planners.
Ultimately, research will always be expensive and the real reason for the bad budget estimations is the political pressure that forces NASA to operate at such a low budget (compared to other industries with less or equal economical return). A pressure that is created by politicians who are funded by these private companies and not acting in the interest of the public.