Summary says it all. I know there's no consensus among the developers. But meanwhile a self-updating program is become a thing that very much users simply expect as an integrated feature.
Possibly a nice integration could be, that the plugin Managers update pane also provides a core update as the very first list item...
I think it would make more sense for the "About jEdit" dialog to have a way for Updater to add a panel to it.
This way, if updater is installed, it can show a button saying "update available" or a message "you are up to date" depending on updater's settings.
I don't think it makes sense to have jEdit itself delivered through the plugin manager update pane.
This is slightly unrelated, but it would also be nice if Updater was provided as a default plugin for Windows and maybe Mac. On Linux the package manager should take care of core updates, but Windows doesn't have one and most Mac users don't use theirs.