I would wait until Wednesday and contact American Express during regular business hours (EDT), and ask them to re-open the account and restore the points. You did not ask to close the account, you pushed a button to speak with someone.
Although probably the best option at this point, it may be a bit of a hard sell. For anyone hearing such an explanation, it's going to sound a lot more like someone cancelling 3 weeks ago, recently discovering that the 30-day rule doesn't apply to them since they don't have another card, and then claiming to never have intended to cancel. And a lot less like someone that accidentally canceled their card 3 weeks ago, but didn't realize it until now. Assuming the machine falsely interpreted your request to cancel a card, what happened next? I find it hard to believe that it didn't at least confirm once that that was your intention, and after that, said something like "Okay, I've canceled your account." What did it say at the time that made you think that something other than your card was cancelled? And, what were you actually calling about that obviously didn't happen, and why didn't you call back in order to get the original task done?