OP asked "Would it look okay to paint the ceiling and crown molding in a flat Pure White?"

No. It will look like a mistake.

I say yes, it will be nice. Ceilings should be painted flat, no glare from natural light or lamp light.

If it were mine, I would paint the whole ceiling sheetrock, including the crown and cofferd trim boards the same flat white.

I think the OP is asking about the ceiling molding and the crown molding, not the ceiling itself.

Typically, the trim on the ceiling would match the trim on the walls in both color and sheen.

Ceiling is flat paint. Walls are matte or eggshell (sometimes satin, or semi gloss, but, not often anymore), trim is satin or semi gloss. And yes, you can use pure white.