Do birth certificates automatically get mailed to you after taking baby home?

Had my baby in August and vividly remember filling out the paperwork for his birth certificate. The hospital team took it and said they would take care of filling it for us. We (foolishly?) assumed that meant the notarized certificate would come in the mail.

It’s November 1st and we still don’t have a certificate and now he’s uncovered by my insurance until January. We weren’t able to add him to my policy without his birth certificate as proof of the life event within 31 days of birth and my HR center has a hard 31 day deadline.

Are we inept first time parents and we’re required to order the actual certificate or should a copy have come to us by now? We’re in Jackson county, Missouri. TIA!