Apache2's mod_rewrite enabled and allowed, but not working

Question

I have a website running on a machine for several months. I need to deploy it on a new machine, so I am doing all the configuration, including the one of Apache2.

Since I am using Slim framework . I need to redirect all my requests to the index.php. For this I am using Apache2's mod_rewrite, but it doesn't work.

I tried to do everything like I did on the previous machine, but I might miss something.

Here is my .conf file, /etc/apache2/sites-enabled/vigisolement-dev-slim.conf :

<VirtualHost *:80>
  ServerName msdi.dev.nfc-interactive.fr
  DocumentRoot /var/www/vigisolement-dev-slim-RUNNING/public
  <Directory /var/www/vigisolement-dev-slim/RUNNING/public>
    AllowOverride All
    Order Allow,Deny
    Allow from All

    <IfModule mod_rewrite.c>
      RewriteEngine On
      RewriteCond %{REQUEST_FILENAME} !-f
      RewriteRule ^ index.php [QSA,L]
    </IfModule>
  </Directory>

  # These lines redirect to an otherweb service, I don't think this to be linked
  # with my problem (I have the same results if I comment them)
  ProxyPreserveHost On
  ProxyPass /japis http://127.0.0.1:8080/apis_vigisolement_java_dev/japis
  ProxyPassReverse /japis http://127.0.0.1:8080/apis_vigisolement_java_dev/japis

  ErrorLog /var/www/vigisolement-dev-slim-RUNNING/logs/apache2_error.log
  CustomLog /var/www/vigisolement-dev-slim-RUNNING/logs/apache2_access.log combined
  LogLevel info rewrite:trace8
</VirtualHost>

My website architecture looks like this :

/var/www/vigisolement-dev-slim-RUNNING/
  |-- public/
  |     |-- index.php
  |     |-- images/
  |     |     |-- example.jpg
  |     |-- (some uninteresting stuff, like javascripts and CSS)
  |-- src/
        |-- (all my sources, my Slim's code)

The behaviour I want :

• If I request for an existing file (like public/images/example.jpg) : deliver it.
• If I request any other file : redirect the request to public/index.php so it can do all my Slim framemork stuff.

The behaviour I have :

• When I request for an existing file (like public/images/example.jpg) : it does deliver it. (Nice !)
• When I request any other file : Error 404 (Not nice.)

I already did a sudo a2endmod rewrite and a sudo service apache2 restart.

Do you have any idea about what I could have miss ? Did I forgot to do something ?

Note that I am using Apache/2.4.29 on an Ubuntu 18.04.2 AWS machine.