I don’t do portraiture, but product photography. These days everything I do is top down flat lay, however, when I was using a backdrop, I got a set of cheap light stands with a cross bar off Amazon and then clamped upholstery fabric to act as the backdrop. The upholstery fabric is much thicker, which means its wrinkles less and is more opaque.
Upholstery fabric is thicker than most fabric you would use for garments. It is used for furniture, like couches, so it needs to be pretty durable. It is a bit more expensive though, so YMMV depending on how expensive your local fabric store is.
As for doing flay lay, that is just what works best for the types of products I am shooting these days. It is kind of nice in that backgrounds are super easy. I usually shoot against a vinyl, patterned backdrop on the table (I got mine from Ink and Elm). One thing it does limit though is the types of props I can use. Lots of stuff looks great in a picture…but not necessarily from the top down.