Nope, it's because the Cyrillic users also are Eastern Orthodox slavs, it was invented specifically for the Slavic language somewhere in Bulgaria, St Cyril then used it to convert the other slavic tribes and give them a free script( for the bible and for other purposes )
Latin is for those converted by RCC, only exception is the Romanians who are EO, perhaps theirs is a romance language that's why.
Greek/Georgian/Armenian were pretty much civilized before their conversion and the latter two had some Persian influence, so perhaps from there their scripts are derived, Greek one comes from phoenician script.
Mongorians and 5 stans are because of USSR influence yes, you are correct.
Fun fact, the status of Brahmi script( father of all our Indic scripts ) as being derived from Phoenician is now disputed, so if it's conclusively proven as not derived, it means it is 100% made in India script