Hi Alex Davis.
Thank you for reaching out here in the Microsoft Community.
In your description, your child is unable to log in to the Family Safety feature via a one-time code.
I note that you've tried not only in Family Safety, but also in your browser to access your GitHub and Microsoft accounts, and I appreciate the frustration and frustration that the current inability to authenticate with a one-time code is causing you and your child.
Since in your attempts, the problem is not only with family safety but also with other areas such as Microsoft account, Xbox, then the issue is most likely coming from the security verification of your son's Microsoft account.
You can refer to the steps below and contact our online support staff at Manage Account Security, who will be able to provide you with more targeted assistance in the area of security codes not working.
Please click on the link below and sign in to your Microsoft account at the top right, you can try this separately using your Microsoft account, or your son's Microsoft account separately.
https://support.microsoft.com/en-us/contactus
1 Click on this Windows icon

2 Click on “Get Family Support” on the left hand side.

3 Type “account” in the search box and click on “Get Help” at the bottom.

4 Then click on “Contact Support” at the bottom of the screen.

5 Select “Windows” for Products and Services, select “Manage Account Security” for the category, and click the Confirm button.

6 Then choose to chat with a support agent in a web browser

Fill in your email address and a staff member will contact you.
If you see that the current chat is not available, please try again at a time that meets the requirements.
In addition, the family safety feature is a feature that is part of Microsoft 365, so if you are experiencing any other issues regarding family safety, you can go back to the previous link and select the Microsoft 365 icon

Then again, select Get Family Version Support and then type Family Safety in the search box, and the subsequent steps are basically the same as what was mentioned earlier.
This can be done by contacting the online support staff for the family safety feature. They will be able to provide you with more targeted help in this area.
It would be an honor to provide you with some advice and I hope my efforts will make things a little easier for you.
Thank you for your understanding and support.
Best regards,
Zev - MSFT | Microsoft Community Support Specialist