Funnily enough, when incorporating, "to make profit" is usually not part of the stated mission, usually it's something along the lines of "to make chess software"

Those missions are generally for employee alignment, recruiting, customer facing marketing. “Making money” doesn’t work for employees that don’t share ownership of the business and are paid at market rate (IOW as low as the market will bear)

Not really, they are part of the charter granted by government and they bind the company in its legal capacities.

Companies are de jure entities, they exist not de facto, but by registration with the state. For many reasons, you or others cannot go to courts and claim different things about a company as a separate entity from its owners without previously having declared to the state and courts that such a company exists and what the rules of the company were.

The mission of a company are thus part of the rules defined for a company at creation, such that if an owner made its riches in oil prospecting, but also had a company whose mission was software development, then other partners of the second company, or creditors in case of bankruptcy, would have no claim to riches that came from oil prospecting. For example.

Obviously money is important. We live in a capitalist world. It lets an org invest in future, greater service to its mission. But it's not WHY a business exists. It's just a measure. It's sort of like saying that you exist to breathe oxygen or pump blood. It's necessary and important, but suggesting that's your mission would be a little reductive.

There’s a reason boards study financials and hide them from employees

The reasons are because it's their job to ensure the organization's future, and financials are the specifics of how the company plans to do that, and because providing financial information to more people than necessary increases the risk that it leaks to competitors and inside traders.