Funnily, you don't know or omit the details yourself.
Russian Empire didn't give the conquered nations the alphabet, but USSR did, as part of supporting local nationalists (surprize!). And it first gave them the Latin script.
Secondly, using different scripts for the same language isn't hard. Serbs use both Cyrillic and Latin interchangeably, and many people used Latin traslit in computers and phones when their codepages weren't available yet, and it wasn't a big problem. It takes you at most 2 weeks to learn Arabic script without knowing the language, and with own language of slightly older version, it's even easier.
You also suggest Arabic is their "proper" language, but abjad is not suitable for Turkic languages -- there vowels are significant, and many more than the 3 Arabic vowel diacritics. They had actually Turkic runes instead. Why don't you bash Arabic too?
What about Germanic peoples? Was switching to Latin from their runes an evil oppression?
It is military force and administration, that set school curriculum, use a certain script, and teach an edited history. Not the Cyrillic.
The Latin alphabet and some local alphabets were allowed for some years after the formation of the Soviet Union, but eventually during the thirties Stalin has started the Cyrillisation by force of most of the Soviet republics. Any opponents were deported to forced labor in Siberia or killed.
After the dissolution of the Soviet Union, most of the former non-Slavic members have abandoned the Cyrillic alphabet previously forced upon them.
> Russian Empire didn't give the conquered nations the alphabet, but USSR did, as part of supporting local nationalists.
Many languages of Russia got their alphabets already in the late nineteenth century or around the 1906 rebellion. If you look at publications then in Mari, Chuvash, Ossetic, etc. the Cyrillic orthography already has most of the special characters that were used in the Soviet era. (Moreover, many of these languages never had a Latin-alphabet phase.)
But in the USSR, official doctrine required crediting the Bolsheviks with the development of minority-language writing, and it became taboo to mention all the pre-1917 developments. Only around the time of glasnost and perestroika was this era revisited in Soviet scholarship, but many ordinary Russians remained unaware they had been taught a myth.
Your claims elsewhere here about Uzbek are out of date. I have traveled extensively in UZ and, as an OSM mapper, I am constantly looking at signage. There is exceedingly little Cyrillic left in most of the country. So little that when one spots it, it seems a bit of a novelty.