Do the Jungian Cognitive Functions/ Processes really exist?
The research done by Dardio Nardi (the researcher you're referring to) clearly shows that something is there. Although, he would have no way of knowing as to whether his samples were of the particular type, or if what's being demonstrated really demonstrates use of that particular cognitive function. E.g doing something logical doesn't necessarily mean you're 'using' Ti.

I do believe we have innate, instinctive preferences and can classify them into types as such, but I truly doubt that Jung's work would be 100% accurate. Even Jung was hesitant to be exacting in his understanding of the psyche - which lead him to leave a lot of holes in his model.

Myers and Briggs filled those holes with many assumptions - which is why the functions are a far better application to finding somebody's 'type'.

So yes, I would say that they do exist. But probably not exactly in the way they were described by Jung. Hopefully people begin to think critically about this, then maybe we can actually see some results.