Thanks a lot for providing a working example. I reproduced it and yes works with that server block. Then I tried to change until I discovered the following. Here is my server block:

server {
listen 443 ssl;
listen 8080;
server_name xxx;
ssl_certificate /etc/letsencrypt/live/xxx/fullchain.pem;
ssl_certificate_key /etc/letsencrypt/live/xxx/privkey.pem;
root /usr/local/nginx/conf/websites/xxx;
autoindex on;
types {
application/x-ms-application application;
}
}

When I access that server using http, Content-Type is returned as defined, when accessing it via https, Content-Type is missing. Strange, isn´t it?
Btw. I am using nginx 1.8.0.