Yes, that's totally unsafe. Still, you have a good point:
...no more unsafe than the 31 year old Perelli’s I’ve been riding on
Indeed! As long as it's a little difficult to pull the bicycle tire's bead over the side of the rim, you've got a reasonable fit. It's still pretty unsafe, but I agree that I'd rather ride a properly-installed, new, wrong tire than a rotting tire in unknown condition.