It really doesn’t matter whether it’s shared hosting, a dedicated server, or a VPS. It also doesn’t matter that you have root.
What matters is that WordPress plugins are designed with the assumption that they’ll be installed into WordPress instances, the overwhelming majority of which are on hosting accounts without root privileges. That means that everything they do – including creating and using directories – assumes only user privileges.
You’re trying to override that and force the plugin to use directories that do require root privileges. Is that possible? Probably. But it’s also atypical; and expecting advice on how to do an atypical installation, from people who aren’t familiar with the plugin, is exceedingly unlikely to get you any good answers.
You’re also trying to use a single directory to house backups that need to be accessed by multiple users, and which is outside any of their /home directories. Can it be done? Sure. But why not just do it the way the plugin is designed?
If you want good answers, I have none. I don’t know the plugin, and I rarely use WordPress.
If you want guesses, my guess would be that the directory needs 0750, with the plugin user as owner and both / all domains using the directory in a group having read and execute permission. I’ll also guess that the backup files inside the directory need 0755 so the user accounts can access the files and initiate restores.
Those are just guesses, and they’re probably wrong.
If you want correct information, you need to consult the plugin documentation and/or search forums devoted to WordPress (or even better, the particular plugin you’re using).
It’s not that people on this board don’t want to help. Everyone here is helpful. They wouldn’t come here if they weren’t. But this is not something we know anything about.