* Storing passwords. You can't store them in plaintext.
* Signing requests (like in the OP's example). What are the alternatives? You can store some kind of authentication in the cookie, which then gets sent with each request. You can use HTTP auth. But in any scenario, you're definitely using SSL, and you're probably using some other kind of crypto on top of that.
* Encrypting cookies. You should store as little as possible in cookies--ideally just a single identifier. But even so, it's very important to make cookies tamper-resistant. You can't, e.g. just have a plaintext cookie that says "userid=42."
So what does the OP propose we do? If this article were making the usual case that we shouldn't implement our own crypto algorithms, I'd agree 100%. But he seems to be taking it a step farther, and saying we shouldn't even use existing crypto libraries, because we'll misuse them. But what are the alternatives?