Does the impending closure of Sweet Briar College portend the end for private womens' colleges that cater to the ultra-rich?

Depends what you mean by catering to the wealthy. I imagine colleges like Smith, Mt. Holyoke and Wellesley that are top notch women’s colleges will continue for some time to come.

Mere “finishing schools” have probably been on the way out for some time.

The more relevant question may be whether college in its current form will continue much longer. The cost of going to college has skyrocketed. At the same time, there has been an explosion of online courses. Arizona State University is completely online. The whole idea of cloistering yourself for four years of college education is starting to seem antiquated. Colleges that cater to the most wealthy will probably be the last to go.