It would if I took this as face value. I think the "independent invention" is debatable which I am sure they are going by the difference in the crops/strains themselves as evidence of such which says nothing about the diffusion of the agricultural methods in the first place.
Well if one insists. Writing is similarly thought to have developed independently in isolated regions but one could also argue diffusion. Anything associated with advanced civilization really. All unique inventions and innovations.
To me it seems more likely it stems from remnants from a once advanced civilization that thrived during the last ice age and got obliterated. So when you say you nothing leads you to conclude such an advanced civilization existed I say look at how after the Younger Dryas Event there are traces left in human knowledge. Fragmented and shattered but enough to reboot in some form or fashion.
If for instance record keeping was part of this lost civilization it could explain how after it got destroyed and pretty much erased, isolated peoples independently developed a written language. If they all had remnants kept in their culture. Different fragments and being isolated under different circumstances, the various peoples took a different route but eventually with several (again) leading to advanced record keeping aka writing.
However if you deny and throw "independent invention" out of the window the conversation ends.