Small absorption units have been available for donkey's years, even gas versions. The problem is that they are grossly inefficient. In a past life, I had one in my office and it consumed many more kWh than a large conventional one would have done

Their big advantage is relative silence.
I guess that a solar version would be feasible, except that a) the sun doesn't shine at night, making the back-up heater a daily requirement and b) placing a fridge in the sun doesn't seem to me to be a wise proposition. I wonder whether consumers would go for a fridge requiring daily maintenance (adding fuel, raking out ashes etc.).
However, a solar absorption aircon unit on a S-facing wall (N. hemisphere) could, I suppose, be potentially feasible, provided that the heat input could be regulated (intolerant of major changes).
When I was a youngster, in the early 30s, we didn't have a fridge. I think my parents bought their first one in 1937. It gave up the ghost in the war, when it couldn't be replaced or repaired, nearly killing the household by a SO2 leak. Ammonia, would be as bad, being highly toxic. Both the ACGIH and NIOSH recommend a 25 ppm TWA concentration with the NIOSH Immediately Dangerous To Life or Health Concentration (IDLH) set at 300 ppm, with pulmonary oedema setting in at lower concentrations. Is this really suitable for household use?
I think a more pragmatic approach may be needed.