Call to undefined function get_current_screen()
This error tells us it can't find a WordPress function that's why I think you're missing some files. The only way to be sure you uploaded all files is to do a manual upgrade.
Will doing a manual upgrade by uploading all the folders except wp-content ensure all the plugins, data and statistics get restored?
All data is stored in your database and not in files and folders.
kind of worried whether I will have to do a manual upgrade for future releases as well.
This depends on your host's settings. See your error logs in your host's control panel or ask them what's the problem.