The running stitch was probably meant to be used as a temporary holder and would be removed, once she had the plate appliqued and quilted in place.
As for finishing, I'd do your Great Aunt the honour of completing it as close to how it would have been done in her day.
I have my Mother's Dresden plate, and the plates are appliqued on with black in a blanket stich. There was no circle put in the centre. Rather the centre edges were turned under, and the background fabric shows thru. Then a quilting design in the centre.