So I called that number and also got transferred 3 times, apparently to the ICVA. I was under the impression we could call the NBME itself and get an answer from the actual agency who processes scores, but the number for them just connects us to ICVA. The woman said it will absolutely be either today, tomorrow, or Friday. She says that the scores aren't delayed at all, it's just because our testing window ended later than some previous years (I still think it's late since it's going on 6 weeks and supposed to be 4). For people who's state applications are due today like Alabama, she said those agencies know when scores are out so the states should be contacting ICVA if there's an issue with deadlines.
My current plan is to just remain patient until Friday, and then get annoyed again if we don't get them by the very end of the day lol.