>> Applications like WordPress or Piwik come with a self-updater, but then you wind up with having to fiddle with the permissions to let the application overwrite itself
Tell me about it! It took me an entire day to figure out how to configure a WordPress install in order to allow it to self-update (without ftp). To do it by only granting owner and/or group write permissions, you have to figure out that you need to modify wp-config.php to define the "FS_METHOD" constant with the value "direct". Without this, WordPress code tries to be super clever with its umask settings, which only makes things worse.
It quite literally took an entire day to set up WordPress to self update. It's easy if you chmod 0777 nuke the entire install, but extremely complicated to set up with acceptable filesystem permissions. WordPress is designed to be sloppily dropped in a webroot, not to be installed by intermediate users who care about security. Well, as "secure" as WordPress can be. And to be honest, the most secure WordPress installation would be incapable of self-updating, as you're granting the web user write access to the entire installation - not just for updates, but for any vulnerability.