This rant would make sense to me if the app was free, but if you're making profit just by repackaging CBC's content they have every right to ask you to stop.
If you're on my lawn selling lemonade, I don't have to "ask" you to stop. If I tell you to stop, you'd better get your butt off, because you're trespassing. Same concept.
Also, it is true the CBC has a right to ask him to stop and that Apple has a right to remove the app at their discretion, but it is not necessarily true that CBC has a legal right to force him to stop. If the app is legal, Apple's policy of removing illegal content is being misapplied to this app.
Edit: Since I received a downvote from a petulant person I will try to explain further. As the writer of this piece explains, his app is basically a paid web browser that only works on part of one site. There is nothing wrong with writing a web browser, charging for a web browser and optimizing it for CBC content.
Now due to CBC's disclaimer on the content they have the right to refuse people from repackaging their content as it abstracts the source of the data. Should the owner (OP) misuse this content in some way there is a chance that CBC become liable. CBC would not sue Mozilla for providing access to the stream within their browser because the stream is not repackaged with non-CBC logos or misrepresentation of their logo. I would wager however that they would pursue a website that embeds the stream and collects revenue from it.
Arguably CBC wouldn't care if he gave it away but since his application would not exist without the CBC Radio content they have a decent grounds for suit against him. The author plays the pity card by trying to sell us the idea he did it for convenience but in reality he aimed to profit from the app. We know this because of his refusal to stop and use his tool privately and his strong attempt to continue profiting from the app dispite legal tension.
The long story short is that the author may have grounds to stand on legally if he got a good enough lawyer to spin his version of the truth but as things stand right now CBC is looking to defend itself legally against any potential liability resulting from 3rd parties profiting from their service.
There is a legal difference between distributing someone's content oneself and linking to it. However, I will grant you that CBC's ToS says:
"CBC/Radio Canada reserves the right to prohibit or refuse to accept any link to the Web site...You agree to remove any link you may have to the Web site upon the request of CBC/Radio Canada."
Of course, that is not the allegation being made. The question is when a web browser crosses from "neutral tool to serve web content legally" to "proprietary tool that serves web content illegally." The for-profit nature of the app has nothing to do with it because there is nothing illegal about charging money for a web browser even though it's not done. The issue is how incompletely the intended web experience is rendered before it is infringement, and I do not believe that stripping out everything from a webpage but the links is infringement so long as the content is legal and the trademark is not being violated.