If you're talking about laying down, then, yeah, many rabbits like to lay on cooler surfaces. But it's different if you're talking about walking about, which I think is what you are meaning.
If the rugs/blankets can be "folded" by her, then they are likely slippery on the tile. No wonder she avoids them. A blanket that can slip about on the tile floor is probably more slippery than the tile itself.
I've had rabbits that could walk just fine on tile (and others who wouldn't step on it). But even the rabbits that can walk on it have no real way to zoom about on it. Those furry paws make it too slick for any real traction.
Perhaps you could consider getting some type of decent size area rug that cannot slide on the tile (with some type of rubber or traction on the underside that holds it in place). This would at least allow the opportunity for zoomies.