ATSAME70Q21 at 1.8V -- cannot program flash memory
Turns out that the problem was neither the 1.8V, nor the rev. A vs. rev. B aspect.
I had not programmed the GPNVM bits — bit 1 specifies whether to boot from ROM (factory default, bit = 0) or from flash.
I find it strange that programming the flash would fail (the flash remains with all 0xFF's) just because the GPNVM bit does not indicate &quot;boot from flash&quot;, but that seems to be the case:  as soon as I wrote 0x042 to the GPNVM bits, I could then program the flash.
The program still won't run, and I invite anyone stumbling into this answer to follow the Atmel Communities forum post if you're experiencing something similar (however, at the time of this answer, there has been no solution posted there).
[EDIT/UPDATE:]
The reason why the program was not executing was the PB12 pin!  Its alternate function is ERASE chip.  If at power-up or reset that pin reads a logic 1, then the chip is erased.  In the previous version, that pin was connected to an LED (in positive logic, so that the LED was playing the role of a pull-down).  In the current version, that pin was connected directly to an FPGA IO pin — although I can program that pin in the FPGA with a pull-down, at power-up the FPGA has not loaded its configuration, and it could be that it has a &quot;hardware default&quot; of pull-up;  in any case, with the pin floating there may have been all sorts of race conditions.
Once I connected (via a wire-patch on the board) PB12 to GND, the board now works like a charm!
[END EDIT/UPDATE]