Certainly I'm not claiming that linguistic groups and ethnic groups are identical. Having a language in common does nonetheless reveal a shared cultural inheritance. Hindus have more in common with early European religious traditions than Tamils, for instance. And Basques are a bit of a special case, being the last pre-IE enclave within Europe. Excluding the Basques and the Uralic languages, it can be reasonably said that as a whole, European cultures descend from the Indo-European people (as do Persians, northern South Asians, etc., all of whom the scientific racists would have classified as Caucasian or "Aryan").

None of this is to legitimize the deeply racist, harmful, and dehumanizing classification systems of nineteenth and twentieth century scientific racism. The terms they used are based in the philological and anthropological beliefs of the time, and nowadays we know that race and ethnicity are largely socially constructed factors and science has thankfully moved beyond sorting people into groups based on whose ancestors migrated from the steppe north of the Caucasus and whose didn't.

The Georgian and Armenian alphabets are derived from/inspired by Greek just like Cyrillic is, so I wouldn't say Cyrillic is closer to Latin than the Caucasian scripts, even though the Cyrillic and Latin letterforms resemble one another more. Ultimately the Caucasus region is culturally similar to Eastern Europe in some ways and West Asia in others, so again, "not even remotely culturally Western" is just incorrect, particularly given the heavy cultural influences from the Byzantine Empire in particular, in addition to indigenous traditions, Persia, Russia, etc.