The Kursk tragedy is a topic for speculation. No one could save the crew. Most were killed immediately. Those who remained lived less than 48 hours after the disaster. The search for the submarine alone took almost 16 hours. During the remaining time, no rescue ship of any country could have approached the place and launched rescue operations.
Another factor is that such operations involve very high tech equipment that is top secret... the last time the Russians asked the west for help... during the Chernobyl melt down, the US Universities were asked for robotic equipment to enable the fires to be fought and the reactor inspected without risking the lives of people... they agreed to help and the US government vetoed that help because they didn't want to hand over robotics technology to and enemy.
Russia had no reason to think anything had changed...
And why would Russia think the west would help, the west repeatedly refused offers from Russia to help during major storms in the US, or wild fires, and even now the west refuses to look at Russian vaccines collectively... individually it is of course different... why are western countries letting thousands of their people die for such pride?