A weakness of the clock idea is that the power could be off multiple short periods; too short to allow the meat to spoil, but enough in sum to make the clock pretty far off.
Battery-backed microcontroller logging all power off events and temperature changes :)
If the battery has run out, then there has been enough long power outages that the meat is probably spoiled. If the battery hasn't run out, you have a nice log of what happened when.
Not that I think its better than the ice solution, but theres certainly techie options that you could build for $10 or so.