Often, a user might bump into permissions errors when trying to install gems on the same Ruby version that came installed on your OS. That version is usually "protected", it's meant to be used for common operations required by your system and you'd be better off avoiding to rely on it for your apps. Technically, you could do some workarounds, like sudoing the commands to force them to run, but it's not recommended to do so as you can end up messing up with basic configs.

The recommended way is to install Ruby through a package manager (such as Mise, as recommended the official docs link you shared in your original message).

By doing so, you'll install a Ruby version that's isolated from your OS and you should then have permissions to manipulate it as you wish.