Radiant floor sellers will tell you that their systems work with any type of floor.
Contractors that do repairs will tell you stories of floors gone bad over radiant heat, like hardwood that warps or laminate that delaminates.
I will tell you that the performance of the system will be affected by the insulation value of the floor covering. Carpet will hold some of the heat out, wasting some energy. But exactly how much, and whether it will be enough to matter, are very specific to each situation, so I can't guess.
Yes, the room can be very cold because of the floor.
Concrete slabs do not insulate well (R-value around 0.6!), and they lose most of their heat along the outside edge of the slab (the ground beneath the slab is a decent insulator). Using a Heat Loss Calculator, I assumed you had a 10x10 room with a completely uninsulated floor (concrete), uninsulated walls (stick built w/drywall), and ceiling (drywall ceiling with an uninsulated attic).
Heat loss was 4050 BTU/hour from the floor, 5440 BTU/hr from the walls, and only 1540 BTU/hr from the ceiling.
So, while your ceiling may be a factor, it's probably not the most major factor. Given that your floor is very cold, you probably don't have much (or any) insulation on that slab.
On the outside of the building, see if you can see the edges of the concrete slab. If you can, then it's not insulated. Buying some one-inch R-5 foam insulation and affixing that to the perimeter of the concrete slab drops the heat loss from 4050 BTU/hour in half to 2050 BTU/hour. Two-inch, R-11 insulation drops the heat loss to only 1050 BTU/hour.
Placing a rug on the floor would likely be insufficient, as carpet only gives an R-value of around 1.3 (and an area rug isn't going to cover the entire floor, anyway).
Edit with more information:
Consider a house built on an uninsulated concrete slab. Insulating the slab can lead to a reduction of 10-20% in total heating costs. Of the heat lost through a concrete slab, up to 80% is lost through the exposed outside edges.
Best Answer
Although finish treatment of old wood flooring might (to some small degree) fill small gaps between planks, this is not the primary purpose (nor will it do that very well).
As you suspect, the primary benefits are aesthetics and protection. As an added benefit, a finished wood floor is also easier to keep clean.