There is an ergonomic element present too. If you stand and let your your arms dangle loosely by your sides, look carefully at the relaxed natural curve of your hands at that moment. It is not the tighter curve of "cupping a tennis ball", nor are the fingers straight or "flat". Instead, that curve is completely natural and neutral. Ergonomically, that is considered the best and safest hand position to use on the keyboard and is highly effective in artistic playing too. It's what I always use.
I think Rachfan expresses it very well in his discussion about the 'ergonomic element.'
When we walk we do not think of our toes in terms of curved or flat when we need to step over an object. The big muscles guide us. If we want to move something away with our foot we do not flatten our toes. Why should we flatten our fingers to play the piano? By keeping the big muscles in mind, our fingers fall into place more easily as long as the elbows are loose. The fingers then are able to reach where they need to be, with the arm acting as a lever.
For playing fast passages I have been taught by German Diez and Jose Aldaz (former students of Claudio Arrau) to think of the larger muscles as well as the position of the fingers. Thinking of the hand as an extension of the arm, and the fingers as an extension of the wrist and hand, is an approach that taps into the natural way our muscles work. Try dropping the arm wight from the shoulder, keeping the elbows loose and unlocked, and the thumbs relaxed to the point of feeling weightless. One can both rotate the wrist and place the hand over the notes. The wrist should easily rotate like a wheel that has no friction. The fingers are then in place from the wrist which should be as flexible as rubber. Proper wrist rotation enables the fingers to be at the right place at the right time. This usually results in fingers with a degree of curve in them. For me, a consequence of flat fingers is tension and stiffness in the hand muscles. This does not let me 'play into the keys' to produce good a tone. The fingers, wrist, elbows and shoulders all need flexibility to facilitate this action. Relaxed technique enables speed and accuracy. The upper arms control the fingers, and from a neurological standpoint, the weight of the upper arms is what enables the fingers to produce the subtle pianistic nuances needed for an interpretation.
I have taught some people with cerebral palsy or paralysis whose hands and fingers are totally stiff. They cannot be flexible and need a different approach. It is always the big muscles that get their fingers where they need to be. Of course, even that is a challenge for them.