Interesting question. I've seen some stuff in the past on how to replace the screen using the current frame and hardware, but never how to totally replace it. I agree, it's not a real rugged set up. We seldom use ours. Plus, it's possible that a time or two I've had a beer or been otherwise distracted and walked right through it. Not confirming that, just saying that it could happen.
This may be too easy, and not real high tech, but what about sewing a screen that completely covers the entire door area and is secured with hook and loop taping?