Normally these threads on Flyertalk involve the OP having done something wrong but still blaming the airline.

But in this case I have to agree that the agents in MIA were out of line and clearly looking for excuses to deny somebody boarding. The mere fact that they put the OP on the next day's flight for no additional cost makes this obvious.

Also, I have NEVER in my recollection been required to show the payment card when using Avios. Moreover I would completely expect to be able that demonstrating knowledge of the credit card number / access to the account, etc. would be fine.

FWIW you can pay the taxes with Paypal, which does avoid this issue. It's also never a bad idea to book 1 parent with 1 or more kids on one PNR and the other parent with the remaining children on the other PNR. That way at least half of the family could fly as planned if something weird happens.

I would definitely get in touch with AAdvantage because they may be sympathetic and refund the unused awards on Iberia. Although a stressful moment on day 1, you really should have called AA before leaving MIA instead of just abandoning those tickets...