Five of the seven responses you've gotten thus far are from instructors, so it might not be a representative sample of opinion on the subject.
I'm an Instructor... with a slightly different take...
What you have here is a difference between learning to dive locally vs. on vacation... and maybe a step further - between land based diving centers using quarries and those that actually do OW certs via boat.
If I had a student who couldn't dive, regardless of the reason - as a local land-based Instructor, I would simply reschedule the dives for him at a later date. No refund, but he would have an opportunity to complete his training at our next checkout. Everyone is happy. If he were leaving for a trip and woke up sixk on checkout day... I may even re-schedule at the local quarry and make special arrangements to get him done in time for going on vacation. None of this can be done when time is so sensitive. In other words, you took a chance that all would go perfectly - and it did not. It's sort of like a gamble.
I have people call me all the time... and say, I'm leaving for vacation in 3 weeks - I need to be certified and finished before then. I work out a schedule to accomodate them - then they call the day of checkout - the weekend before their trip and say something came up for Saturday - can't make it... what can we do? I tell them their only other option is to complete their certs by referral and they'll have to pay the Instructor whereever they're going... then they want a partial refund... and I'm like - no, sorry. I was more than willing to get you done -we had an agreement.
If you break the agreed schedule... I say no refund. If I break the schedule - I owe you a refund or I need to find a fellow Instructor to fill in for me.
In the case of your 12 year old, he by being sick, broke the schedule. I have little doubt that had your vacation schedule allowed, the center would have gotten him completed and certified - and hopefully at no additional cost. That is what I would have done... but the fact of the matter is - you were either unwilling or unable to do this... in either event, you broke the agreed contract.
Do I think you deserve a refund - No... would some places have given at least a partial refund - Yes... because they're a different kind of business person than the ones that only care about the dollar.
I can't see either is entirely right or wrong.