Nowhere on the internet is this specific problem or a fix for it mentioned, so here goes:
My app contains the following doGet()
and doPost()
POST and GET works perfectly fine for me using the exact same code you posted. My guess is that you are not testing the POST correctly.
Here is my published URL. The code behind is a copy/paste of your sample -
https://script.google.com/macros/s/AKfycbzWZv9WUp7rUtOxZhhwuXPpuNXuvPGpiHyrcYrPeNLiusOWzazo/exec
Test it from http://hurl.it (simple REST tester in a browser) and works without any problem.
Request (make sure to check follow redirects) -
Response (response after the redirect for one time ContentService URL for both GET and POST)