Resolved: ESP8266 works when arduino UNO reset pin is grounded
The RESET of the UNO puts the ATMega328P (the MCU of the UNO) into an electrically inert state when it is connected to GND.  It is not supposed to be running your sketch when it is being held in reset.  It is, roughly speaking, electrically disconnected from everything else on the board.  This allows you to use the UNO as though it were a USB to TTL serial cable.
If you want to be able to run a sketch on the UNO while it uses the ESP8266, you'll need to not hold reset to GND and you'll also, practically speaking, need to choose a different set of pins for communicating with it, as D0 and D1 on the UNO are the same signals used to for the ATMega328P to talk to the host and used for programming the UNO via its bootloader.  On the UNO, you'd instead need to use SoftwareSerial (or something similar) and pins other than D0 and D1.  [Update: see the original poster's own answer containing the guide they ultimately used, making use of SoftwareSerial on pins 10 and 11]
The ESP-01 can run Arduino sketches using the ESP8266 Arduino core and it may make a great deal more sense to not involve the UNO at all, and instead connect the distance sensor more or less directly to the ESP8266.