It's more than just oversimplification. There were a lot of political discussions among Russians (especially among Russian decision-makers), what do they want, how they want to achieve it, who exactly fits that job, etc... Those discussion are usually less public than in America, but often gave wider spectrum of political choices. Before 1999, political candidates for becoming Eltsin's heir were mostly pro-Western liberals (like Stepashin, Kirienko, Hakamada). Ideas of cooperation with America were mainstream there and anti-American ideas were mostly part of political minorities - loud, but not really influencial.
1999 changed almost everything. Russian decision-makers made their choices, and the mainstream changed on "Appeasement doesn't work. We should stop them. Let's start confrontation with West". And the man fit this position was found. It was Putin. Further NATO expansion, as well as illegal and unprovoked aggression against Iraq proved that this opinion was correct.
