You are correct that "run its course" means "to continue until finished". If something "has run its course" then it has already finished.
Often a sick person is told that they must rest and let the illness run its course. This is the meaning used in these sentences. The sense that something is creating change and that the change will stop at some point. This phrase is often applied to ideas. When an idea has run its course the idea is no longer causing meaningful change. The idea has run its course.
Both of these sentences use that same definition. They are saying that an idea has concluded.
Many people believe that feminism has run its course.
This sentence is saying that feminism is no longer causing meaningful change in our society.
It was a wonderful show, but I think this play has run its course.
A show or play is built around an idea. Once that idea is resolved the show is fundamentally concluded. Anything extra can only take away from the meaning of the idea.
Shakespeare's Romeo and Juliet can be taken as an example. The central idea is that two star crossed lovers are separated by political reasons. Once Romeo kills himself the play has ended. It does not really matter what happens after that point since the central idea has concluded. At that point the play has run its course.