This isn't reddit a specific problem... They're just the latest flavor of this news cycle. Other companies have gone through this same exercise... most recently twitter but also Google and Yahoo.
At the end of the day, the API costs money to run. If it's "free" (it should never be free) and someone builds and app off the free API they're essentially getting compute (and everything associated with running the service) for free. Maybe not a huge deal sometimes for small traffic sites but once sites reach a certain size it's a beefy chunk of money and that's extra not good if the company isn't not rolling around in profits. And, I ?think? Reddit is still losing money.
On top of that, if their api allows video/pics and the media doesn't have moderation then porn money will suck up a huge amount of bandwidth costs (traffic you can't generally monetize regardless if it's via a 3rd party through the api or served first party).
IMHO, it's a mountain made out of a molehill. If there's some amazing 3rd party app/feature the community wants then just pitch to have it added to reddit app.