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.