If you don't do full length, it doesn't need to be on the ground or get stepped upon.
Paper is more forgiving than the $10 sheet if you want smooth and featureless. If you can't notice wrinkles and the sheet is out of focus, you will probably be fine. But if it gets swept to the side and has ripples in it like a curtain, they will show up.
All the studio images in my Instagram are taken with the same gray cloth background. It has a texture painted on it (not prominent but not invisible) and I often have folds in it like a curtain, such as here:
As you will notice, most look like different colors. I have not changed the texture but I have changed the color.
Obviously I am using the texture.
Background Paper is similar to cloth in that it has a fine texture (cloth texture can vary). Paper reflects light in a more consistent way (it isn't "reflective" like plastic, it is more muted). Cloth can reflect light less consistently across its surface due to the weave, the materials used to make it, how the cloth itself is unevenly positioned unless it is stretched taught and how consistently opaque it it. Neutral gray paper is about as neutral of a surface as you will get.
Do you need that level of consistency? I have no idea. It's kind of like asking how important color consistency is in strobes. If you are doing catalog products, it will make your life easier. For my portrait work, the incredible color consistency level my strobes achieve is probably a waste .