Having inherited an allotment that had been carpeted, I'd never choose to do it. Our allotment had mature currant and gooseberry bushes that had been planted through carpet around 10 years ago. The carpet had disintegrated
enough that the weeds were rampant through it (
bindweed and couch grass, nice), but you couldn't remove it, when you tried to it just fell to pieces, definitely not natural materials, I think it was mostly the glue and possibly latex backing? Just crumbled into the soil, it will be there forever I suspect. The only times I've contemplated using carpets are on top of plastic sheet mulch (so when it's done its job I can just roll the whole thing up and throw it away without losing bits into the soil) or as a very temporary cover for a
compost heap, etc.
What about
cardboard? You can get huge amounts of free cardboard - from bike shops and appliance stores are good because they are BIG pieces - then you dont have to worry about removing it, it will just rot down.