Why is the flu shot free? Economics, actually. A few years back there was a study done at the public health level of the costs of flu, particularly including hospitalizations for the two major risk groups of children and the elderly. The net result of that study was that mass vaccination against influenza saved tens of thousands of lives, but more importantly prevented hundreds of thousands of trips to acute care centers.

Recall that flu is a virus. While it can be treated with antivirals like oseltamivir or baloxavir, it's not terribly common for mild-moderate cases. In fact, they don't routinely test for it unless a person is hospitalized and it would make a difference in their care. Typically, treatment is "go home, rest, eat some chicken soup, wash your hands and don't spread it." That's cheap. Tamiflu and Xofluza aren't cheap ($300-500/course), if they're used, but they're a hell of a lot cheaper than an ER visit or an inpatient hospital bed.

By preventing hospitalizations, the flu vaccines save the system a HUGE amount of money. More importantly to health insurers is that they save the health insurers huge amounts of money. So, a business case was made to major insurers that they would benefit financially from covering the flu shot at no cost. And that is why the flu shot is "free" (included in the premium) on any health plan in America.