Call me nuts, but I often do check any alternative flights on the day to see what would have happened. Anyway, I figure if I tried to fly "day of", something would happen, but when I fly day before, it would have been okay to wait.
Only once did we not get to port when expected, but arrived next morning after having to stay overnight in CLT. That was in Jan 2000 and we had kids in tow as well. We made it from CLT to FLL no problem the next day and bags were all good, so it worked out, but is certainly stressful. I don't recall what would have happened had we flown "day of", but we might have made it not much later in the day.
Still, I wouldn't think of attempting it in winter nor overseas, especially with airport snafus in recent times. GL to OP.