Your description of growth from the center of a flower bloom sounds like quite the phenomenon. Hope you took pictures of that. Perhaps it would help the understanding process. It reminds me of when all my pink periwinkles started blooming green. Pale green petals on every stem. I thought it an extra terrestrial pee-ed in them or something silly. Turns out it was just the nutrient levels mixed with the type of soil they were growing in.
Are you sure it is a coneflower? Not a type of helianthus or rudbeckia?
... Let's define asexual propagation.
Asexual propagation is used to maintain selections of known identity and quality and includes such techniques as division, air-layering, grafting and cuttings. Asexual propagation creates plants that are genetically identical to the parent plant.
Sexual propagation is a natural process resulting in a parent plant forming seeds that create offspring that are not genetically identical to the parent plant, as in asexual propagation.
Both types of propagation have positive attributes. Asexual propagation allows you to reproduce or clone the parent plant exactly. This is especially useful when the parent plant has desirable characteristics such as a white cone flower or hybrid appearances.
Asexual propagation preserves the characteristics of the parent plant. The plants produced by asexual propagation will also flower faster than those produced by sexual propagation because plants grown from seed need to pass through a juvenile period before they flower.
Asexually propagated plants are mature when they are propagated and begin to flower immediately.
Sexual propagation has several benefits as well. Growing from seed is cheap and easy. Growing plants from seed produce offspring which are not genetically identical to the parent; therefore, the propagated plant will be genetically diverse from the parent plant. Such as a white cone flower reverting back to mauve color. I've heard plenty of success stories in asexual propagation involving coneflower/echinacea. But, it takes patience. Division would be your best bet.