I agree with Pat’s comment about focus of the airstream being the key that unlocks good sound from many flutes. Also agree that playing flutes in higher keys is a good way to find focus, and get muscle memory of what focus feels like. So are overtones.
What follows is uninformed by any knowledge of the technical aspects of embouchure design, but comes from messing around with flutes over the years. Some flutes are easier to blow, and easier to get a good sound out of with an undeveloped embouchure. I think the reason is that something to do with the design of the embouchure somehow results in larger sweet spot, which means that it performs pretty well even without that focused airstream. A poor analogy, but one sees the same thing in golf club design-optimizing for forgivingness.
Similarly, there are some flutes that are more difficult to get a good sound from with an undeveloped embouchure. I think the reason is that they have a narrower sweet spot that is harder to find, or is only found if the airstream is focused.
I think the transition to getting a decent sound of nearly any flute is a sign of acquiring a focused embouchure, which improves the sound from most embouchure designs.
Hugh