Magento 2 running on Ubuntu 20.04 with Nginx returns Bad Gateway Error code 502

Question

I have a fresh install of Magento 2.4.4-p2 on Ubuntu 20.04. The PHP version is 8.1.13 and the nginx version is 1.18.0 (Ubuntu). I can see and access the home page and the following pages without 502 bad gateway error:

https://example.com/sales/guest/form/
https://example.com/catalogsearch/advanced/

Everything else including the admin page and any activity on the above pages such as a search query returns bad gateway error 502. Inside /etc/nginx/sites-available I have example.com virtual host with the following configuration:

upstream fastcgi_backend {
   server  unix:/run/php/php8.1-fpm.sock;
}
server {
  listen 80;
  server_name example.com;
  set $MAGE_ROOT /var/www/example.com/html;
  include /var/www/example.com/html/nginx.conf.sample;
  client_max_body_size 2M;
  listen 443 ssl http2;
  listen [::]:443 ssl http2;
  ssl_certificate         /etc/ssl/cert.pem;
  ssl_certificate_key     /etc/ssl/key.pem;
  access_log /var/log/nginx/magento.access;
  error_log /var/log/nginx/magento.error;
}

I followed the documentation and adjusted the below in /etc/php/8.1/fpm/php.ini :

memory_limit = 2G
max_execution_time = 1800
zlib.output_compression = On

Usually, this issue is caused by an incorrect Nginx configuration file. But this is what Magento 2 documentation stated. How can I fix this? - The error log is clean.

The content of include /var/www/example.com/html/nginx.conf.sample; is here.