Linen actually doesn’t take to large scale mechanization very well. It causes the fibers to break into shorter pieces more often, which makes the final fabric rougher and less sturdy. Machine-woven linen also tends to be more loosely woven, which is again less sturdy.

Machines certainly helped some amount, but cotton got a way bigger boost from industrialization. That’s why cotton is so much cheaper than linen today, especially high quality linen.