You can do this. I recently switched hosting companies and had to move over a few sites.

You do, however, need to copy some of the old folders in the original install (hopefully you still have access to them).

You need to copy over the /config, /files, /packages, /blocks (if you have overrides) and /single_pages (if you have overrides).

Once you do a fresh install, copy these files over the new ones, restore your database (including adding a user and assigning the user to the database) and then change these (database name, username, and password) in your copied /config/site.php file.

I believe I've remembered all the necessary steps (from memory). Nonetheless, that should do it. There's NO need to run the install after uploading a fresh copy. Copying the old files and changing the site.php file should restore your site back to the last configuration you had when you made the backup.

Hope this helps!

You also need to copy/move the /themes directory (if you're using a custom theme or one not from the core install).