I'm not a linguist or an expert on mythology, but this theory seems very unlikely to me for several reasons.

The first is that the association with Phoenicia is just a theory by later authors based on the similarity between the word phoinix and the name Phoenicia. It need not have any real basis and doesn't really prove anything about where the idea of the phoenix comes from. The ancients also theorised that the named was due to the fact that the bird was purple (purple dye being named after its place of origin, Phoenicia), but this is similarly just folk etymology. Herodotos claimed he head the story of the phoenix in Egypt, where he was told the bird came from Arabia, bypassing the Phoenicians altogether.

Second, as noted in the book The Myth of the Phoenix (1976) that is cited on the Wikipedia page you linked, the Greek word phoinix is very old. It is attested on 2 Mycenaean documents written in Linear B, the script used in Greece before the adoption of the Phoenician alphabet. In this syllabic script, the word is rendered po-ni-ke, notably not including an F sound but also apparently breaking the link with the other words you've offered as elements of the word phoinix.

Third, in terms of the way Ancient Greek works, the etymology you're suggesting is impossible. Greek works with stems that tend to retain their integrity; the word sophia (wisdom) is present as the stem "soph-" in words like sophoi (wise), sophistês (wise man/teacher), philosophos (lover of wisdom/philosopher), and so on. This word is always reduced to its first syllable, never to its second. It is recognisable by its stem, and not using the stem would make it unrecognisable and therefore unintelligible. In any case, the concept "wisdom of the night" would not translate sophoi nyx but sophia nyktos. Finally, the "-nix" in phoinix is spelled with an iota, the Greek equivalent of the letter i. But Nyx is spelled with an upsilon, which is represented in the Roman alphabet with either y or u. This is an altogether different vowel, and the original pronunciation of Nyx would be closer to "nooks" than "nix".

I hope this helps!