As I pointed out in a sibling comment, I think Netflix actually has the same problem. There is a significant presence of long-running shows that people love to watch on Netflix: Friends, Breaking Bad, Modern Family, How I Met Your Mother, etc.. A big favorite "It's Always Sunny In Philadelphia" just left the catalog - presumably due to the acquisition of FX by Disney - and "Archer" will likely soon follow for the same reason.
A lot of those are not new and people still watch them (either to catch up on a missed cult hit or to rewatch their favorites), and the will continue to do so for the next 10-20 years. As long as that's the case Netflix also has to continue to make deals with the existing rights holders or substantially shift their core offering.