'''Important: Apple has a non-publicized internal replacement program for this issue '''
As posted above, I began having "Touch Unresponsive" issues immediately after updating to iOS 11.1 on my iPhone 6 Plus with original screen. The phone would be unresponsive to touch for 10-30 seconds and then would suddenly "catch up" even as background functions continued. Putting the phone to sleep and waking would often allow me to get it to start responding again, but only temporarily and not 100% of the time.
What did not work: 1) updating incrementally to 11.2.1 as updates became available. 2) doing a restore from encrypted backup to 11.2.2.
What did work: I made a Genius Bar appointment (Apple Cherry Creek, Denver) and eventually talked to tech Tyler who after running diagnostics told me that:
1. Despite coinciding with the iOS 11 update, it was a hardware issue
2. He had seen the same issue on phones running iOS 10
3. Because my phone was out of warranty, Apple had a replacement policy that would give me a new replacement phone (same model) for the standard screen replacement cost of $149. (And because this would also get me a new battery which I was planning to pay $30 for, my cost was effectively $119. Presumably if you're under warranty your cost will be zero).
There may be other factors going on, perhaps it's a factory installed screen that is having trouble with iOS 1 but either way, I came away with a new replacement phone that worked with a 90-day warranty and a new battery for $149. Since the 6 Plus is still plenty of phone for me and I expect to keep it for at least another year or two, this was far better than paying $800 for a new phone.