Of course, Soviet Union did exactly that under Stalin (in Mongolia as well).
However, they would not, and did not, try that on the main historically Christian nationalities that already had established alphabetic writing traditions, namely the Baltic and Christian Caucasian nations. It would have been difficult and fairly pointless, I suppose.
In earlier times, if you want to Russify, you do not offer them a variant of the Russian writing system. You either force Russian (the language, not just the script) down their throats, or, if you can't (and, in Finland and Poland, the Tsars just couldn't) you let them writing their language in whatever script they like to. Also, the vast majority of Finns, Poles, ethnic Russians and pretty much anyone else in the Empire were illiterate regardless of language (possible exceptions including the Baltic Germans, who would of course be literate in German). So again, little point.