MCM couch webbing replacement  

Jute
I used jute webbing.

I wouldn't use jute
It doesn't have any stretch. The stuff you see there is Pirelli rubber webbing. It can be purchasd (google fagas strap).
But there is a cheaper alternative. If this were a very high end piece I'd recommend original pirelli replacement rubber, but I think you want something that will make the sofa comfortable like it was meant to be, and not cost more than it is worth. Go to the upholstery specialty store, and ask for the black elastic webbing with a red stripe down the middle. Or any elastic uphostery webbing for seats. It will stretch about 10%.
Your sofa looks like it has an unusual strap attachment. Normally there is a metal clip that gets fixed to the end of the strap and clips into the slot on the chair. But on your chair there is a wooden strip that has a nail at each strap attachment point. I am guessing the nail goes through the strap. But if the straps do unclip then I am wrong.