If you read the article again, notice how the fast majority of those millions of dollars have gone to private companies who failed to deliver working systems. Each of those companies passed multiple rounds of competitive bidding, too. The government staff processing things by hand can at least point to a ton of successfully processed requests rather than a complete write-off.

Outside of shallow libertarian magical thinking, privatization isn't a panacea. Anyone who has worked for or with a large corporation has stories which are at least this bad — the main difference being that you rarely about about the failures unless it leads to a major lawsuit. The usual argument is that you aren't required to do business with those companies but … just try to live without insurance, telecommunications, cars, etc. Markets allow you to select the least-bad option but it often takes regulation or changing the problem to make significant improvements. Before someone mentions Google: consider how happy you'd be if their QA-in-production / no live support mentality was applied to your retirement checks rather than your free email…

There are two real problems here: the first is the well-known fact that we're still figuring out how to deal with large-scale IT projects, which affects everyone, and the fact that government agencies are significantly blocked from doing things the right way. For the last couple decades, politicians have “shrunk” government by giving jobs to the private sector rather than hiring lazy government workers. What this actually means is that instead of hiring the same person directly they're hired at a significant markup – equal or greater pay plus the overhead costs of all of the contracting company's staff and profits – and all of the work is pushed into a procurement process which forces you into the worst possible form of waterfall.

You're not the first person to notice that things aren't efficient – and I'm certain that staff at OPM would agree, given how many of them were quoted saying just that. The solution is easy – allow them to hire good people in house at market rates and give them the needed support / autonomy – but that's politically dicey. This and your proposed solution are both likely to fail as long as Congress has a sizable contingent of people who benefit from problems, even those created by their policies, and everyone receives donations from companies which do business with the government. This also won't magically go away with privatization, as anyone who has seen large deals go to the company which bought the best lunch can tell you, but at least in the case of the government the details are public knowledge.