I have personally narrowed this problem down to the Broadcom Touch IC chip. I'm thinking when technicians go to plug that long connector in they tend to not have it aligned properly and short it out somehow causing the grey bar and random touch loss.
Your options here are the following:
1. Restore the phone to OEM condition with a broken screen or not and take it into Apple playing dumb so they will replace it.
2. Send the device to a specialist who can replace the chip for you...as low as $30-$40 on eBay.
3. Order the chip and do it yourself if you are capable.