I don't believe so. Some people feel that it improves the edge, I have never noticed a difference. YMMV.
IMO (which is backed up by an antique barber manual I read) the cloth component is on antique strops as a sacrificial cleaning component to prevent a build-up of soap and gunk on the expensive leather side. You have to remember, a lot of homes and barber shops in the old days did not have hot running water.
As long as your razor is kept clean and dry I dont think you will miss it.
I mainly use mine If I am stropping with pastes to make 100% sure none of them end up on my leather. Also some people report an improvement stropping with cloth after honing with coticules. I'm still experimenting there...