YTA. I would probably have had the same or similar reaction and I would have ended up apologizing. You said it: it was a mistake. Key point: nothing bad happened except for your argument

I always plan for the worst and hope for the best. I have high expectations of others. I get let down a lot. You can hold on to the anger and dwell on a situation that didn’t have a bad outcome OR you can look forward and try to avoid a similar situation in the future. If you really think you need to keep it going and push harder, wait until emotions die down before doing so. At the end of the day, it sounds like she loved the kid but slipped up. I would just try to avoid coming at the conversation all high and mighty.

Also, talking to the kid when you’re really talking to someone else and saying it loud enough for the other person to hear it is just being passive aggressive and is not the best way to start off