It's not ideal that you have to use a mishmash of services to get what you want when you want it. But there are multiple delivery mechanisms for a lot of popular content.
How does this mean that TV is broken? Because you can't get instant access to every show ever created? That's not a TV problem, that's a you problem. That says nothing about TV. It does point to a weakness in distribution, but there are a variety of ways to get legally content these days: over the air, cable, satellite, online streaming (Hulu, Netflix), online purchase (Amazon, iTunes), buy or rent a DVD of a season of a show, etc... all of which can be played on a traditional TV.
If you want to make an argument, argue that distribution is broken. TV as a content production industry is still doing okay.
BTW - that was a good episode of the West Wing. While it was all done in flashbacks, you get to see the history of how people came to work in the West Wing. Donna's character particularly got more interesting after that episode. But you probably don't want to watch just that episode, it was the second half of a two-parter.
...the content production industry is not doing okay, because it's working "for hire" for people who are almost exclusively interested in advertising revenue.
If the people who produce TV shows had good contracts, they could also sell me their content directly - with no middle men. Or at least, different middle men. There's no technological reason they couldn't.
I agree that you're right that it's a distribution problem.
But there's a supply of video, and a demand for video, and people standing in the way of that market. The people standing in the way are eventually going to go away.
...that's my belief, anyway. Until then, when I have money and want to watch a show, but can't, TV is broken.
I cannot give them my money, to watch it. I could wait a couple days for Netflix to mail me the DVD, but by then the impulse might pass.
TV is broken.