I think I have the same problem (unless I have not understood you).
One of my userlists has the search facility showing all the time (viewing from the frontend). All the other lists that I have created - chosen from the dropdown menu - have a "Search Users" link which reveals the search facility with a slide down action only when you click it.
It was the default list but the problem persists with that particular list even if I make a different one the default, or unpublish then re-publish it.