Removing the furler
drum, foil and head roller assembly from the unstepped mast, to store inside the boat would be PITA, IMHO. But this would deter theft and protect it from the damaging effects of freezing water and snow.
For all the years prior to ownership of my current boat, the boat was stored indoors after unstepping both masts. The masts, with furler attached, were placed on a rack with others, the entire rack then shrinkwrapped. A great system, but too expensive for my budget.
We, along with 90% of the 400 sailboats at my marina, keep masts stepped, although some in-water sailboats will remove the spars for a tighter winter cover fit. The masts which remain on the boats of course, have the furlers exposed. Seems to be the norm with not much concern.
We do shrinkwrap our boat though, over a custom metal frame. The furler drum is therefore protected under plastic, along with the head roller, which can be lowered with the jib halyard