So I've used Apple support three times via their chat feature.
Given that I had already tried a DFU restore, the first had me try "Reset All Settings" on the iPhone. After a bit of time I concluded that didn't work.
Next step was to backup the phone (full backup) on iTunes and then do an erase and setup as a new phone. This puts it so it only has Apple software and has the effect of determining if any of the non-Apple software is the cause. After a few days I concluded it still had the problem.
Apple tech support said this then means it has a hardware problem. The next chat person connected to my phone remotely and after I authorized in the Analytics section of Privacy (it goes by different names -- advertising and analytics I think), did a hardware diagnostic. Turned up clean. I highly recommend using the free Apple chat for "touchscreen not working as expected" to see if they will do this diagnostic remotely and tell you if you have a hardware problem. Each of us might have different issues going on so you might get a different result than I did.
But that person said the next step was to go to an Apple store. Did that. The analytics the tech did was about the same as they do remotely with a couple other smaller checks. Came up clean again. BUT! He said it must have a hardware problem. Could be the logic board or the touchscreen so a new phone is the best bet. I was left with these choices: 1) suffer 2) replace the phone for a few hundred bucks 3) upgrade to a new phone via the iPhone upgrade program (which is the price of the phone divided by 24 as monthly payments for 24 months -- no interest!). I decided the guy was probably right. I'd dropped the phone too many times or caused an issue that day I got caught in the rain and it got too much moisture under my jacket.
What this means then is: millions of people have iPhone 6 phones (and 5s etc.) and Apple is still exchanging bad phones like mine with the same model. But only some people have the problem. So it is not the software. There is no bug. There is, however, something in 10.3.1 and 10.3.2 that reveals a hardware problem and instead of giving a coherent message, it gives funny behavior like most computer hardware problems do. I think it is getting stuck in an diagnostic routine as it periodically scans the bad hardware. So we can say Apple caused the problem, but the point is the problem does not exist on phones that do not have a hardware problem. So we can all be mad our hardware problem didn't cause issues until 10.3.1, but we had an underlying hardware problem.
Anyway, that's my logic. Be persuaded or stay mad/sad/frustrated. Or figure out the real cause and solution if you think mine is not the one :-)
I've moved on to a new phone. Good luck all. And really do try the free Apple chat tech support. But most likely at the end of it you are looking at repair or replace. :-(