For starters, in the West, Greek affixes were used in scholarship, where it was felt they were more nuanced than what Latin had to offer. Suffixes to express ethnicity were felt to be a less rarefied domain, and English and Latin between them had it covered.
For seconds, Greek differentiated between suffixes denoting ethnicity, and adjectival suffixes. –ikos was only the latter. So a vase might be Athēnaïkos, but Thucydides could only ever be Athēnaios. Just as he was a Hellene, and not a Hellenic.
That’s why when the –ic suffix is used against countries, as OP noted, it is used as a scholarly specialist term, rather than as an ethnic term, and it is used as a convenient way to differentiate a major language from its superfamily. Germanic vs German, Turkic vs Turkish.
This is terribly inconvenient for Greek, in which Germanikos and Tourkikos are merely the adjectives for German, Turkish. The former is accordingly rendered as Teutonikos instead, but such synonyms are not usually available. The only real solution for the latter is to call them Tourkogeneis Glosses, Turkogenous languages — that is, languages that originated from (small-t) Turks.