I just spend two hours on the EA support chat talking to two different agents about exactly this. Quite a lot of the time spent trying to get them to understand that it wasn't an issue with passwords or the child or my adult accounts themselves but launching and signing into the actual game on the Switch.
Finally got them to acknowledge that their troubleshooting and online resources are not fit for purpose when it comes to child accounts. No resolution, a vague 'we've noted the issue' and a 'have a look at our forums'. Great,
Next time I'm faking the birth year so I can create a teen account with an email and an account i can actually link to my adult account and manage via Family Management.