While polls do show Russians no longer trust the West, they also show Russians very much want to cooperate with it. Chechnya is a localized issue focused around the specific perceived threat of terrorism, not a symptom of naked Russian aggression. One must recall the events underlying public support for the war. Late last summer, Chechen warlords invaded the loyal but highly combustible Russian region of Dagestan, declaring their aim to establish an Islamic government there. Rightly or wrongly, Russians believe that the terrorism originated in Chechnya.
Stories also flood Russia about kidnapping rings and a Slavic slave trade based in Chechnya.
The public, therefore, supports the war because it wants to counter what it sees as a serious and very immediate threat to its security, not because a majority harbors a desire to reestablish imperial rule over a wayward province. All this certainly does not excuse the reckless, vengeful killing that Russian troops have perpetrated on the hapless civilians of Chechnya. But it also means Russians are unlikely to support similar aggression in other foreign policy spheres in the absence of horrific episodes like apartment bombings to strike terror in their hearts. Putin's support is broader than the Chechen War. Russians clearly want a strong leader, capable of bringing order to their tragically unpredictable lives.