I just found reason - breakages only happen if overview is option to place on to place on the left is enabled. Once this option is off, all breakages disappear - all keys work, session file list is saved, documents close. No more those failed assertions.
Could be that it depends on the order, in which widgets are created. Could be similar case as with "second editor" pane option, where second pane is as primitive as possible in features, comparing to primary editor pane.