So WWI historian, does this expression even exist?

As far as I know, the expression did not appear at the time nor was it associated with a particular battle. The Sabaton song is probably where it originated from.

which is about the battle of Passchendaele and the enormous amount of casualties (and more generally about WWI).

Regarding "the price of a mile" as a measure of military success or lack thereof, battles and campaigns in both world wars were often quite costly, and regardless of how much ground was taken. German 11th Army suffered 50% casualties in the course of the very successful Gorlice-Tarnow Offensive, while the Russians suffered close to 2 million casualties in the course of 1915, half of them POWs. Yet despite the heavy losses and the territorial gains made by the Central Powers, Russia remained in the war for two more years. Similarly, the Russians suffered 1.5 million casualties in the Brusilov Offensive, and the Austro-Hungarians over 1 million, and the Russians made impressive gains. But Austria-Hungary remained in the war until late October 1918, and the Russians collapsed in 1917. German forces in Operation Barbarossa experienced a casualty rate per day similar to that of the German 1st and 2nd Armies on the Somme in 1916, with almost a third of the invasion force being killed, wounded, missing or taken prisoner by the year's end. Allied casualties per day in Normandy were slightly higher than they were for the British during 3rd Ypres in 1917. Had the Normandy campaign gone on twice as long as it did, 152 as opposed to 76, take away the German prisoners taken in the Falaise Pocket and the casualties suffered by both sides would have been similar to those of the Franco-British and German forces on the Somme (the Somme lasted 141 days).