Origin of the phrase Sica
Referring to a Sword Sica without mentioning the Slavic connotation is a serious misinformation for the sake of propagating a nationalistically oriented approach to the Albanian origin for whatever reason.
There are clear references to the Proto-Slavic, which is not attested by the connotations in a written form, but make total sense geographically and ethnologically. https://en.wikipedia.org/wiki/Proto-Slavic
Proto-Albanian, on another case, is simply a stub, something that's not possible given the historic influence and geographic positions of the Slavic-spoken people in the region. https://en.wikipedia.org/wiki/Proto-Albanian_language
There is no attested connection between the Albanians and Illyrians, and the phrase "Thika" derives from the phrase "Sica", which in turn got introduced to the Latin languages thanks to the Roman historians who wrote about the sword used by a different culture. Sica is NOT a Roman sword. Sica did not originate in the Latin language as evidenced by the historic reports.
Proclaiming that it is LATIN in origin is a serious misconception and a fallacy.
Again, Albanian language is not attested before the 15th century in any written form. Illyrian language is not attested in the history in any written literature work. The "Proto-Albanian" as a concept is ludicrous at best, especially since the phrase Sica got introduced to the Latin languages. In Albanian language it is a loanword from the Latin, but it is not originally an Albanian blade nor is the name of Albanian origin. Furthermore, the very phrase Sica would be the same on Albanian language, which is not the case.
https://en.wikipedia.org/wiki/Sica
De Vaan did not study any Slavic language nor is he a good source on the origin of the sword's name since he only took the side of Latin phrases and sentences without mentioning any other Balkan people who had a different culture such as Dacians, Thracians and Illyrians. It is not a question of his reputation, but of his objectivity.
I'm confused. You claim, "[t]here is no attested connection between the Albanians and Illyrians," yet the w:Proto-Albanian language and w:Albanian language articles both mention a likely derivation from Illyrian. Are you claiming that both of these articles are incorrect? Or are you trying to claim that the lack of attestation of a connection is proof that there is no connection?
You also claim, "the phrase 'Thika' derives from the phrase 'Sica'", but Albanian diachronic sound patterns do not seem to bear this out. A quick survey of the Albanian lemmata here on Wiktionary shows that ⟨th⟩, /θ/ derives most commonly from PIE /k/, with occasional instances of shifts from alternative Albanian terms with initial ⟨f⟩ instead (not surprising, phonetically; observe the opposite /θ/ → /ɸ/ shift in certain dialects of English). The only clear cases where Albanian initial /θ/ comes from an earlier /s/ phoneme were theks (“to say”) from Proto-Indo-European *sokʷ-o-, and thi (“pig”) from Proto-Indo-European *suHs -- but in both cases, the vowels are back vowels, not the front /i/ vowel in thikë.
Meanwhile, many modern Albanian terms starting with ⟨th⟩, /θ/ are proposed to derive from earlier forms starting with ⟨ts⟩, /t͡s/. Latin lacked this phoneme, making a shift from /t͡s/ to /s/ a likely adaptation for borrowed terms -- more likely than the proposed /s/ → /t͡s/ shift for a borrowing from Latin into Albanian.
Your insistence on a possible Slavic origin for the Latin term sica is also puzzling, as the term is attested in Roman sources two millenia old, whereas the Slavs didn't migrate west until the 5th and 6th centuries. Unless you are also positing that the Slavs were either time travelers or common tourist visitors to the Illyrian coast, any Slavic derivation is geographically unlikely.
De Vaan aside, your case is not convincing.