Covering the door sill would be fantastic.
I think you may be able to keep it flush and avoid it becoming a trip hazard if you add shape-strengthening ribs and/or have the liner follow the shape of the door sill beyond the flat area (because adding bends is a way of adding strength to the shape). Do you know what I mean?
I couldn't find an example for this material, but it is done quite commonly to metal parts like this:
View attachment 47583
View attachment 47584
I mention because in my experience sticking velcro on carpets hasn't been great.