Russians need a travel visa to go to any Western country and most of the world. Some EU countries are banned Russians from entering; the US is not issuing travel visas in Russia anymore.
US is actually quite good on offering entry to refugees from Russia. At least 30,000 people from Russia entered US through Mexico and requested asylum in US and many got it. The problem is that it's only option for basically rich citizens of Russia because whole process is expensive, hard and quite dangerous.
EU is much closer, but it does nothing. Putins regime could've lost 30-50% of it's high-skilled workforce if EU or UK just made it easier to immigrate. E.g literally 100,000s of Russian IT workforce left due to war and political situation, but getting actual work visas is hard process and outside of country of citizenship it's only gets harder if not impossible.
But honestly west can't even help Ukraine efficiently. How can one expect EU to actually do anything to cripple Russia economy...
There are a lot of political immigrants from Russia as well as people who trying to avoid being drawn into army. And for people who left Russia back in 2022 it's just basically impossible to get any visas anywhere simply because you can't apply for one outside of Russia without having some other residency permit that' impossible to get in Georgia / Turkey and many other countries.
EU still provide visas to tons of people who continue to live in Russia and pay taxes in Russia, but dont give any visas to people who left and dont support Putins regime.
And a tourist visa is hard to get even with a residence permit. The consuls (rightly) see you as an immigration hazard. After all, you've already moved countries one time, who's to say you won't repeat the trick?
Meanwhile in Moscow, you have a good chance to get a 5-year visa from France.
Fun fact: the exact same phenomenon was being ridiculed by the White Russians, back in the 1920s. European countries were suspecting them of being Bolshevik, yet the actual Bolsheviks could come just fine.
Now, of course tourist visas are not really relevant for emigration, but it's an example of the attitude shown towards us.