Why didn't the WWI Allies attack through the Vosges Mountains after 1915?

The Flanders front was flat. The typical WWI soldier got standard clothes, a rifle, a knife and a shovel and was almost ready for action in such a terrain. A really broken line here means an enemy is free to move as much he is able to, logistics more often than not being the limiting factor.

The Vosges are quite steep, have almost half a year of Russian Winter on them, and both sides fortified their lines after the Franco-Prussian war of 1870. The French had an Iron Line (Barrière de fer) built immediately behind the mountains and the Germans built bunkers and block houses along the highest points of the whole range. At the start of the war they had trenches that were already made of brick. Breaking a line in such terrain means you face just another steep hill next.

Note that the Wikipedia states the French fortifications were already obsolete at the start of the war. That is an anachronistic assessment by "bullet points". The Germans always had considerable difficulty in dealing with such fortifications. The most famous of these forts had a rather known name: Verdun.

In such a terrain you need mountain troops. Significantly more expensive to train and equip, much smaller numbers available. Logistics a nightmare. Just compare the Austrian and Italian experience in the Alps.

That said, one personal account of a snapshot in time does not capture the reality of 4 years of war.

The Hartmannsweilerkopf did change hands 4 times alone in one campaign – with 30 000 dead – and was called the Mountain of Hell or the Man-eater-hill. There is one of four French memorials commemorating the war nearby.

Collier's New Encyclopedia (1921)/Vosges Mountains
VOSGES MOUNTAINS, a range of hills in the E. of France, bounding the valley of the Rhine on the W. and parallel with the Schwarzwald in Baden. They are connected in the N. with the Hardt in Rhenish Bavaria, in the S. W. with the Faucilles, the Côte-d'Or, and the Cévennes, and in the S. they unite with offsets from the Jura. The summits are usually rounded, hence called ballons, and are covered with a rich green turf, on which for six months of the year large herds of cattle graze. Their sides are clothed with forests of fir, oak, and beech. Their highest summits range from 3,000 to over 4,000 feet, culminating in Ballon de Gebweiler, 4,700 feet above the sea. In those mountains was carried on severe and almost persistent fighting during the World War.

That means, yes,

an army on foot was better off charging into interlocking fields of machine gun fire west of Nancy at Verdun, the Somme, and Passchendaele

because in the Vosges mountains, where they did try the same as everywhere, an army on foot would be charging into interlocking fields of machine gun fire as well, but additionally in much worse terrain, climate, and the enemy usually holding the high ground.

Epilogue:
Less known now, just like most of the action in and around the Vosges mountains, is the story of Sergeant Stubby, whose story provides a backdrop for the conditions there:

Sergeant Stubby: Heavy winter snows in the Vosges Mountains were holding back French supply lines; mules and horses couldn’t breach the impasse to move artillery and ammunition. Allan managed to transport, in secret, more than 400 sled dogs from Alaska to Quebec, where he and the dogs boarded a cargo ship bound for France. […]

One of the more prominent participants in fighting in the Vosges mountains had the name Harry Truman:

He saw his first action in August 1918, amid the mud and mire of the Vosges mountain range in Alsace-Lorraine, firing an artillery barrage and being fired on in return. The captain stood his ground. Many of his men did not. He cursed them for it, and won their respect.