She signed uo for this when she applied for a gc: she agreed to live in the U.S.
There were plenty of opportunities for her to file I-407 after she abandoned her LPR status. She then could have attempted to get ESTA, and if that failed, got in the (long) line for a B visa.
TBH, except for LPRs who are were forced as minors to leave by their parents, I never understand why a non citizen goes through all that trouble to get a gc, and then tosses a privilege that millions are waiting to get.
Nothing, and that includes ailing family, would cause me to abandon my LPR privilege. And I managed the end games of my father, mother, and brother, cremated them, settled their estates, etc while still spending most of my time in the U.S.
She cannot be denied entry.
What is the evidence that she cannot get ESTA ever?