It's always puzzled me how objects with properties like rest mass and charge (and color) could really be geometric points. Is this just a fiction needed for the math to work? Could quantum field theory work with spherical electrons and quarks?
No, it is not fiction.
It is the present conclusion from a large number of data fitted by quantum field theory calculations that hypothesize that the standard model particles are point like. No other models today can describe the totality, or almost totality, of the present data.