I have caught kite lines with bare hands before but only when they were moving slowly. For a fast moving kite, grabbing the lines with unprotected hands would likely create a severe cut or worse. Even with gloves on the line could cut right through them. There are these cut resistant kevlar fiber gloves out now that might help but perhaps not enough.
Perhaps a rule of thumb could be:
1. Grab the kite and work to the center of the leading edge with the kite inverted or to a wing tip to disable the kite.
2. If you miss the kite and it is moving slowly and if you have hand protection you can try to grab one line. Even with protection you may still get cut if you miss-judge or a gust hits.
3. If the kite is moving fast, don't try to grab a line. Yelling at the top of your lungs to bystanders to stay out of the way may help.
It is complicated, but creating a runaway kite generates more potential complications and issues than this. I think focusing on avoiding runaway kites deserves a good deal of attention. This discussion indicates that if you miss a fast moving kite, you might be best to let it fly on and cause whatever damages and injuries that may come of it. Trying to help and losing some fingers for your effort won't aid anyone.
I am going to add a bunch of accounts to the KSI tonight. Many deal with irregular leash function. We are far from predictable safety and fully reliable safety devices in this sport at this point. The best and most reliable safety device we have is exercising good judgment to avoid emergency situations.