Ok i worked ticket/gate for years. I’ve been out of that game for about a year so take with a grain of salt.

You should try to fix this over the phone. I see you did already try, worth asking again.

In any case, corrections CAN be possible, but if there is anything else wrong, like a misspelled name then we are getting into territory of it being a different person. Name changes are not allowed.

The agents at check in should be able to fix a simple DOB error. It happens and corrections happen. If you are near the airport and can visit a check in desk, you can take care of this days or weeks in advance. Or just go good and early on the day of departure.