Russia actually did try to come close to EU(more specifically the West, which includes the US). This was before the US invaded Afghanistan. Russia started co-operation with the NATO on information sharing regarding the terrorist activities in the AfPak region. And also provided some logistical support. In return Putin not only wanted to be part of NATO, but also wanted close collaboration with the West on all matters from finance, tech, media etc. Basically Putin was trying to integrate Russia with the West. Unfortunately, that co-operation was ended once the invasion of Afghanistan started. Putin might have felt that he was used. He went back and started a "modern-style" warfare against the West i.e. cybercrime, social media propoganda, taking down of Russian-origin collaborators of the West etc. This was very well presented in a documentary I saw few years ago on the analysis of Russia's political maneuvering against the West. Unfortunately I am not able to recall the name of that documentary.