I'm curious how, without server admin privileges, WP accomplishes this automated tweaking of the software one's server is running
That's easy, they use FTP to upload files with the permissions of the account owner, just like we do to manually tweak the server while it's running. So WP really doesn't have access to all your servers unless they stored and transmitted your FTP account info when you submit it, which it doesn't.
Plus, it's all source code, thousands of people look at it daily, we would know right away if some funny business was going on.
However, that would be a great exploit, to make a rogue plugin that started capturing the FTP user/pass when people try to update their site or install new plugins.