The principal advantage of a flat wire is there is a little less drag.
Possibly it is also less likely to break than a stranded wire.
I don't know whether it is more or less likely to kink than ordinary stranded wire.

SailsEtc sells the flat stainless steel wire and it does not seem too expensive, so why not use it for shrouds, jib topping lift and perhaps backstay?

I don't understand why you think it will alter the stiffness of the mast.