It is not the Russian way, but the proper way of making war.
The NATO doctrines are basically based on German teaching, or to put it correct on what the former Nazis made up about their doctrines.
The fact is that the Nazis lost WWII and so does NATO in every war ever since.
Soviet and Russian doctrine was about the same as NATO. Indeed, it was even more focused on deep operations and combined arms.
This was worked out between Marshal Tukhachevskys crew and the German delegations in the 1920s-30s, when they were cooperating on tank design and doctrine.
That enhanced blitzkrieg doctrine was the Soviet standard throughout the Cold War. That’s what the Soviet group of forces in Germany was structured to do.
The Russian army went to an even more extreme doctrine, with combined arms and independent operation down to battalions. Those were the BTGs that entered Ukraine in 2022.
You could see them attempting to implement their doctrine. But they were too incompetent to do it. Imagine Guderian running out of gas two days after crossing the Meuse in May 1940, and German logistics failing to get him any for a week.