Alright, so here's the solution:
Return the Application Claim named (Email Addresses) from your Azure AD B2C User Flow for Sign up and Sign in.
Do this by checking the appropriate check box under Application Claims:

Answers to the questions I asked earlier are:
- Why is the user presented with this screen when they sign up to a Power Page site with Azure AD B2C?
- Because the token sent from B2C login screen does not contain a claim of Email Address (or Email Addresses) as it is called in Azure AD B2C User Flow Application Claims. As the user's email address is missing from the token, the user has to enter it manually.
- Why is the user asked to provide their email once more, when they just had provided it in the screen before?
- When the necessary Application Claim is returned from the B2C User Flow, the user will not be asked to provide their email again.
- How can this be avoided?
- As described in the solution above.