Come to think of it, I don't think there's a good reason to bother with the textile-braided cords on clothes irons (etc.) anymore (besides tradition), as there has been a technique to modify PVC (cross-linking the molecular structure via carefully controlled irradiation) not to melt for many years now. Hook-up wire with such insulation (abbreviated as XLPVC) is actually fairly common for internal wiring of equipment (UL AWM style 1430 or similar); I'm quite sure they wouldn't be using it if it wasn't cost-effective, especially in the current economy (well, it's more of a false economy than anything else to tell the truth, but that's another story). It even has the designation "V4" (where "V" is standard PVC, "V2" is high-temperature PVC for up to 90C, and "V3" is PVC for cold conditions) under the harmonised coding system (so H05VV-F for example would become H05V4V4-F), or so I read.
I'm pretty sure a cord using it would be more reliable than the braided type, at any rate (given that the braided cords are particularly prone to kinking - over-tensioning the conductors and in some versions also revealing the inner coloured insulators - and at least some braid types will eventually fray from normal use pretty much regardless of how well-treated). And guess what? Some versions of the braided cords feature XLPVC inner insulators anyway!
