This is an interesting problem. As far as I can tell, most commercial programs use some version of Photoshop's "dust and scratch removal filter", about which I can find nothing about its implementation. It's probably some despeckling technique, one method of which uses what's called a Lee filter:
http://angeljohnsy.blogspot.com/2014/08/lee-filter.html
However, most modern despeckling methods use wavelets, and I don't know if Photoshop includes a wavelet transform.
From a naive perspective, I'd imagine that a "star" is a white blob (possibly with blurred edges) which is significantly brighter than its surroundings. This means that you should be able to remove stars by setting a threshold level, and removing all blobs for which their mean value and the mean value of their surrounds differ by at least that threshold level. (This may have to be done in an adaptive way, changing the threshold across the image, maybe by checking the variance of the surrounding neighbourhood: this is looking like a Wiener filter). Having removed the blobs, the holes can be filled in by using an interpolation technique.
I'll have a a go at this (if time permits) over the next week or so, and see what I get.