Yes there are open source softwares that once completed are efficacious in solving the problems they were designed to solve. The point yeukhon was making was the act of open sourcing a project alone doesn't somehow inherently solve problems in its development. At least in this case not the two enumerated problems he sees with the healthcare.gov project.
When Wikileaks gets their hands on government secrets and puts it on a website, they haven't made those secrets 'open source'. That word doesn't even make sense in that context.
Even if I get my hands on the source code of Windows and put it in a GitHub repository, I still haven't made Windows open source.
That is a different petition to the one that has been posted, which states that the reason for open sourcing the code is "so we may help fix any found issues".
The "let's see what we paid for" argument is an appeal to the more general proposition that there should be public scrutiny of public money. There already exist mechanisms for accessing taxpayer-funded content held by government (including source code) under FOI and other legislation. Unless there is some inadequacy in the operation of those laws, I think they answer the point.
We need to do those things you mentioned in addition to open sourcing the code. Its our code let's see what we paid for.