Which architectural element let Gothic cathedrals push walls aside for glass?

The flying buttress. A stone vault pushes outwards as well as down, and a Romanesque church resisted that with enormous walls and small windows. A buttress takes the outward thrust across an arch to a pier standing clear of the building, so the wall between the piers carries almost nothing and can be replaced with glass.

The consequence is theological as much as structural. Abbot Suger, rebuilding Saint-Denis in the 1140s, wanted light as a means of approaching the divine, and the whole Gothic project follows: walls dissolving into coloured glass, and interiors that get taller and thinner for two centuries as builders learn how far they can push it.

They pushed too far at Beauvais, where the choir was raised higher than anything before it and collapsed in 1284, and the transept tower came down in 1573; the cathedral was never finished and is currently held up by internal bracing. Most of the great cathedrals took a century or more, span several changes of style, and were built by masons who had no calculations at all - only proportional rules, models, and the accumulated evidence of what had stayed up.