So… I should have checked before making this post, but it was the usual suspect : selinux.
It was preventing php-fpm from establishing a connexion on port 443 (ence no outbound traffic).
Everything is working now.
In cas you have the same issue, I used these commands:
sudo ausearch -c 'php-fpm' --raw | audit2allow -M my-phpfpm
sudo semodule -i my-phpfpm.pp