The central approach is arguably "make the West do it for us".
USA have been downsizing its consular services in Russia for around five years, both directly and by provoking Russia doing so as a reciprocity (such as, force Russian consulate in SF to be closed, Russia responds by closing one in Ekaterinburg (or was it St. P?). This also made life of Russian expats in the USA worse as they could not reissue their documents.
EU was for some time more lenient and issued a lot of tourist visas in Russia (there were no visa waiver program even before 2014), but after COVID-19 travel decreased immensely, compounded by the fact that EU refused to recognize Russian vaccination shots. That led to most of EU visas that Russians held expiring and not being renewed.
After the war has started, the USA has closed MasterCard/Visa cards and PayPal payments for Russians. USA and ECB also prohibited shipping physical cash to Russia. That means many Russian citizens do not have the ability to take funds with them abroad even if they have those. They have also obviously severed air traffic between Russia and EU/US - meaning that Russians usually have to go through some 3rd country such as Turkey.
Granted, some Russians do indeed move to places such as Georgia, Turkey and UAE. But they don't consider these countries as a final destination and so they may return to Russia after some time - many already do.
In general, neither the USA nor EU really wanted Russian immigrants that much. In the earliest years of 90s one could travel to EU/USA and get a refugee-like status, but that was shelved quite quickly. Germany wanted Russian citizens of german or jewish origin but had no desire to accept russians. As Russian classics said (self-translation on the side),
Я завтра снова утром синим Tomorrow early chilly morning
Пойду евреев провожать, I'm gonna tell the Jews goodbye
Бегут евреи из России, For Jews are fleeing Russia proper
А русским некуда бежать... While Russians having no way out...
in that Russians did not have any obvious emigration prospects and don't have them now. This was not true for some scientists in 90s and IT specialists in 00s onwards, but most of Russians don't have marketable skills and usually don't speak any languages besides Russian fluently.