Removing index.php from CodeIgniter 3 URLs

I am attempting to remove the index.php from my CodeIgniter 3 URLs. Currently, all of my URLs look like this:

http://example.com/index.php/Controller/function

And I want them to look like this:

http://example.com/Controller/function

I have followed along on the following article and Stack Overflow post, yet none of the answers I've found have worked for me:

• https://www.codeigniter.com/userguide3/general/urls.html
• How to remove "index.php" in codeigniter's path

My directory structure is set up as follows:

• /var/www/example
  • /application
    • (standard CodeIgniter application files... controllers, models, views, configs)
  • /public
    • /css
    • /js
    • /fonts
    • /images
    • index.php
    • .htaccess
  • /system
    • I haven't touched anything in this directory
  • .htaccess

Contents of /var/www/example/.htaccess:

<IfModule mod_rewrite.c>
    RewriteEngine on
    RewriteRule  ^$ public/    [L]
    RewriteRule  (.*) public/$1 [L]
</IfModule>

Contents of /var/www/example/public/.htaccess:

<IfModule mod_rewrite.c>
    RewriteEngine On
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteRule ^(.*)$ index.php/$1 [QSA,L]
</IfModule>

Contents of /var/www/example/application/config/config.php (trimmed to what I've changed based on prior reading):

$config['base_url'] = '';
$config['index_page'] = '';
$config['uri_protocol'] = 'REQUEST_URI'; /* have tried AUTO as well */

For my Apache server, mod-rewrite has been enabled via:

sudo a2enmod rewrite
sudo service apache2 restart

My virtual host file for this site is:

<VirtualHost *:80>
    ServerAdmin [email protected]
    ServerName example.com
    ServerAlias www.example.com
    DocumentRoot /var/www/example/public

    ErrorLog ${APACHE_LOG_DIR}/error.log
    CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>