JFTR, I didn't mean to suggest it should be a per-document filter. Only that it would maybe create the most comfortable solution but also with the most effort needed.
Getting back to @techee's first idea of this feature: when we want tojust jump to the first match of a symbol, it might be enough to enable GTK's treeview search feature (where a little text input field pos up automagically and filters the tree, https://docs.gtk.org/gtk3/method.TreeView.set_enable_search.html). Then we don't need a permanent visible search field and probably even less code. At the price thata search feature enabled by just typing in a list is less intuitive and might not be recognised by all users. I didn't try it yet, it's just a thought.
In case someone wants to try, there might be signal handlers on the symbol list to switch focus to the editing widget on any keypress, at least it feels like this.