But then that tech department would become government itself, and would that be better?

Maybe, maybe not.

Though rare, there _are_ private success stories, such as healthcare.gov 2.0 [1] and Palantir, though the latter gets in hot water for some tangential reasons. But maybe figure out how these happened, and scale that.

[1] https://www.theatlantic.com/technology/archive/2015/07/the-s...