Dont install wordpress from cPanel.
If you already have stuff in your root folder then you can install the new site on another subdomain.
Create the subdomain in Cpanel (newsite.example.com or whatever) then copy the package and installer file into that subdomain folder.
Having created the new database, go to newsite.example.com/installer.php, enter the database details and off you go.
When you have resaved the permalinks, go to the settings page and change the bits (both of them) where it says newsite.example.com to the main domain name example.com.
Now, in cPanel go to domain manager and set your example.com domain to point at the subdomain newsite.example.com.
Done.
I’ve done this for hundreds of sites and its fine every time.