The area type textbox only exists as part of a type object. As soon as the object is converted to paths it is no longer a type object, and the textbox ceases to exist. Two options that I can think of are, first, as you suggest, to draw a rectangle and set it behind the (former) type or, second, to go to the Appearance panel, add a fill to the grouped letter shapes, move the new fill behind the Contents, and apply Effect > Convert to Shape > Rectangle, and set the parameters.
Peter