But that's not relevant. Loanwords, especially and above all loanwards related to animals and plants, have been happening for thousands of years even when the language itself already has a native term to a similar subspecies or even to the same being. That doesn't even seem to be the case of chestnuts. The cultivation of chestnuts in Europe apparently dates to around 2,000 BC and the variants of the plant seem to have come from Turkey. It was also a very common food staple only in Southern Europe, beginning in the Balkans. Thus a Proto-Greek origin of the word is nothing but unsurprising. There is not only historic evidence for "chestnut, castaña, kastánea" being only loanwords, there are also solid linguistic bases. Following normal sound changes since PIE times, Greek "kastánea" should not give us English "chest[nut]" nor Latin "castanea", almost exactly the same word. This is enough for us to exclude the possibility of this word being very ancient in IE languages of different branches.
so you are telling that becomes was common in Balkans
has also same etymology in Iranic languages?
in fact as you said it may originated in Turkey,
that is good point for caucasian origin of IE
besides what about the tree that is in PIE as ptel-ela
Tilion πτελεα
what about oak tree, did that came from Balkans to West Europe?
what about Ulmus tree,
ww do not speak about tree like
Orange κιτρον that was brought by Alexander
which Strangely or accidently although import from East has different names
we speak about common European trees ok?
so instead of finding a cheery picked even for one tree, better search the forest.
in fact that commonality was the main theme for Anatolian hypothesis
but was the metal onomatology that shook it and ram it down,
cause if I follow that Logic, as you describe it above
THEN IT IS MOST EASY THAT WORD KA-KO copper as the computer
might be from fast expand of metalurgy than from IE marching North and West,
PS
this has nothing to do with stupid Nordisism or Baltisism or Meditteranism,
All are IE today, and world is moving so fast so all have equal chance to prove their 'superiority'
if that help some of us to feel better.