A perfect example of a project that will only ever lose money but is very important to the public is the Hanford Nuclear site[0]. The US government discovered recently that the nuclear waste site was leaking radioactive chemicals and threatening to taint the Columbia River water supply, which millions depend on. Cleaning up the radioactive waste will cost tens to hundreds of billions, and the best possible outcome is that everyone still gets clean drinking water. That should be the most desirable outcome, but even if successful, the government won't generate a profit from it, nor should they. Utilities shouldn't be priced to be profitable, they should be priced to be sustaining and accessible to those who need them.
"Impact" can't be tied too strongly to number of people affected because one of the things governments do is take on jobs which must be done but which are unprofitable because too few people need them done to make it into a market. This should be obvious, but it's also obvious that "number of people" is a really easy thing to measure, so, like the drunk looking for her car keys under the streetlight because the light is better, all organizations tend to build metrics on top of the data that's easiest to get at.
Peter Drucker, in the 90s, encouraged businesses to emulate non-profit organizations. He was popularizing case studies which showed some of the most effective, efficient orgs in the world were non-profits.
From wiki:
"The importance of the nonprofit sector,[35] which he calls the third sector (private sector and the Government sector being the first two). Non-Government Organizations (NGOs) play crucial roles in the economies of countries around the world."
https://en.wikipedia.org/wiki/Peter_Drucker#Key_ideas
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I like your use of "metrics". I usually frame market design in terms of "incentives", but I might adopt your frame.
Price discovery and surplus (profit) are not the only possible metrics. Markets can, and have been, designed to optimize pretty much every other conceivable goal as well.
But the thrall of corporatism (aka Freedom Markets™) keeps those alternatives out of public discourse. Anything not optimized for surplus (profit) is dismissed with pejoratives like socialism.