Is the Always keep database open option intended to work on server files as well?
No. What this option really means is “never close this database when the window close button is clicked, even if the database wont have any windows any more.” But that seemed a bit long for a checkbox title. Since there are no window close buttons on the server, this option has no effect on the server.
On the server, if you leave the Close database after each access option unchecked, the server will leave the database open unless your code explicitly closes it. (Note that for single user databases you can also close the database in code, even if the Always keep open option is enabled.)