Problem is that it requires that the app handle all the security aspects of talking to the Internet, because STDIN/STDOUT is an unrestricted protocol. Normally when you write a webapp you have a well-tested parser to validate the HTTP request for you, and you have safe APIs to access parameters, and your templating language auto-escapes dangerous characters for you, and the response goes through middleware to prevent XSRF attacks, and the whole thing is written in a safe language where there's no possibility of buffer overruns.
With this, you have to handle all of that in your application binary. You can certainly do this. However, if you are asking why you need to do this, you are almost certainly not ready to do this.
I agree that this is a pretty useful method of tying software systems together. However, one of the the systems you're tying together is the public Internet, where all sorts of danger lurks. Writing secure parsers is hard. I've written an HTML parser [1], one of the most well-tested ones out there (I spent the better part of a year doing nothing but testing & debugging, and ran literally billions of documents through it), and Google's security team still found 2 bugs in it with about a day's work.