@arri THANK YOU! I lack the reputation needed to upvote your comment, but THANK YOU! I never even thought about ripping that chip out.
Here's my story:
Dec 25: router died. (Merry Christmas! At least it didn't harm too many other people.) Serial console worked; diagnosed a bad eMMC w/ help from TAC. Found an M.2 drive that worked, and reinstalled the OS then restored settings from backup.
March 30: Died again! This time the console's dead. Tried "everything".
Found your post mentioning desoldering it. Found a chip labeled "Kingston" and decided that must be it (but looked up part# to confirm). Attacked the face of the chip with a flamethrower, er, butane utility lighter, while mercilessly gouging and stabbing it with a sharp little screwdriver. Continued punishment until it flew screaming off the PCB, carrying PCB traces with it. Regretted not using more flame and less physical force. Scraped off traces still hanging on, ensured that nothing left on the board was touching anything that it shouldn't.
Powered it up with fire extinguisher nearby (and video camera running in case it went up in a glorious ball of flame and made my son scream as it tried to do to him what I did to it) and...it booted! Replaced heat sink grease (even dabbed some on the RAM chips that were naked before), wadded up a ball of masking tape and set it on top of the battery to replace the retainer clip that we broke earlier, and screwed everything back together.
Back in operation! No data loss this time -- best outcome possible. Lesson learned: A bad eMMC that was "harmless" before can become "harmful" 3 months later!