Allpoint / Moneypass ATM at Madrid–Barajas Airport?

I doubt it, since those networks are mostly US based, with some limited countries included. Your ATM card should still work, just not fee-free. In Spain, most ATMs will charge a fee for use (3 to 7 euro) and nearly all will offer DCC, so be aware of that.

I would buy euros in the US (or wherever you live or are leaving from) before arriving in Spain or Europe in general. I am not used to exorbitant ATM fees here in Sweden (my US credit union Visa card only charges 1%) but Spain was a shocker. Anyway, I would contact my bank and arrange for euros before leaving. I found that in Spain a lot of smaller places are hesitant to take cards because (at least in 2019) the banks charged them hefty fees.

While no guarantee, I did use a Unicaja ATM with no fees and in the past Rural Caja ATMs without a fee. But it does vary, both fee or no fee and the amount of the fee.