The installation wizard lacks details when things don't work out of the box. I finally solved all the issues by searching for the youngest/last modified files, which usually are the logs. In fact within Prestashop directory, there are quite few which do have details.
find /apache/prestashop -type f -printf "%T@ %p\n" | sort -n | cut -d' ' -f -2 | tail -fn5;
In my case, they were mostly caused by php configurations (php.ini), dependencies and file permission.
If you face this type of issues, you've better chances to fix it with direct SSH access and some control on Apache/PHP