What do you think: should all government software be open source?
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Even if an external company makes it, they can add an open source mandate if they want. The US DoD is starting to mandate the usage of open standards for their contractors to increase inter compatibility and ability to extend those systems.
Open source software has some value like making it easier for analysts to find security issues and the act of open sourcing software usually leads organisations to raise the quality because they don't want to be ashamed of the code. Plus imagine the clout gained by a dev who got a bug fix merged in that millions of citizens get to use.
Even if an external company makes it, they can add an open source mandate if they want.
Sure they could, but like I said, the're isn't really any need for it. It being open source doesn't make it more secure or better by default, but it does means that anyone wanting to exploit it just got handed the full codebase to make it easier.