It doesn't need to compete, not really. There are many bodies of government. National governments, local, state wide, and from many different countries. These all need software, often doing more or less the same. If they would pool their resources to pay development of useful software, theoretically it could overcome a tragedy of the commons and create really useful software cheaply. This increases productivity and thus economic growth.

It may compete with private software for a while, but not that much: companies will find a way to add value to existing open source software or create new propositions. Building out the boring and useful foundational stuff collectively will just move the bar on what is exciting and new software, or what are better takes on existing software. Companies will be creatively seeking out ever more complex problems to tackle once the government builds out the basic tooling.

And ideally, that is what private companies should be good at: quick to pivot, creative and innovative problem-solving.

Of course, that requires governments to play nice and enable companies to leverage their tooling too, and - perhaps a bigger problem - take responsibility for competent governance of the most important projects and manage their adoption well.