Hi,
Yes, the thinking is that by going on the BC pill, it will suppress ovulation. No Ovulation, no cyst. However, I don't know if that only applies to "functional " cysts or all kinds, as some cysts are not dependent on ovulation to begin with. There are lots of ladies that have ovarian cysts (like myself) who no longer ovulate and haven't for years.
Also, at 41, many women are still thinking about having kids, so it's really a personal decision..