Thanks for the tip on "personal hotspot" - I gave it a try but unfortunately same issue.
After a reboot, Messages works and the standard "Vol Up + Power Btn" can be used to reset phone. After error occurs, Messages UI is unresponsive (white screen / black screen --> going homescreen / switched focus away and then back again can get it to redraw the last known screen but remains hung / unresponsive to screen presses), and the "Vol Up + Power Btn" can NOT be used to reset phone after error.
Post-error, can still reset phone using the "Vol Up, Vol Down, Hold Power Btn" hard reset process. I'm guessing this all has to do with the kernel level updates on 16.4.1 resulting in some hung process given that the phone is unresponsive to soft reboot in addition to Messages hang. Happy to provide logs for this easily reproducible problem if any developer ends up on this thread (I wish...). Meanwhile I am resetting my iPhone every 15-30 mins so I can respond to work needs.