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The plastic is there as a VAPOR barrier, not a thermal barrier. It protects your insulation from warm, moist inside air getting into your insulation and condensing into water when the temperature changes inside your insulation (the purpose of insulation: to provide a temperature barrier from the outside). That condensation can, in turn, rot out your insulation. The vapor barrier keeps the humidity inside of that barrier. Even though the temperature might pass through it, the denser moisture-vapor will not. 6 mil plastic sheeting is the most common, simplest, and inexpensive way to do so.
As such, DO NOT REMOVE THE VAPOR BARRIER. If you replace the plastic sheeting, make sure that the replacement is specifically RATED and LABELED to serve as a vapor barrier.
Your insulation should do the bulk of the radiant barrier work, though. This outer barrier merely provides draft protection, similar to the way a gore-tex shell provides wind protection from cold. Gore-tex shells usually suffice as a coat unless it gets below freezing even though the shells really do not provide much insulation at all. They suffice because they restrict direct transfer of cold and warm air. They do so by not allowing the cold air to blow through the barrier. Same thing on a house wrap. It prevents the cold air from blowing inside the insulation layer, thus making the insulation do its job of providing a true thermal barrier.
As far as I can tell, there would be no advantage to adding a second thermal barrier inside the insulation. It should be on the outside, not the inside.