How Can I Deactivate The 'More Information Required' Screen After Sign-In?

For break glass accounts, the general recommendation is to have MFA for them that is different from what everyone else is using, and not to disable it completely: https://docs.microsoft.com/en-us/azure/active-directory/roles/security-emergency-access

There are several reasons why you may get prompted to register your security info: https://docs.microsoft.com/en-us/azure/active-directory/authentication/multi-factor-authentication-faq#why-are-my-users-being-prompted-to-register-their-security-information-

And by default, accounts with Global Administrator role must have their MFA registered (read Azure AD Free): https://docs.microsoft.com/en-us/azure/active-directory/authentication/concept-mfa-licensing#available-versions-of-azure-ad-multi-factor-authentication

My guess that you have SSPR enabled, or the break glass account is a GA in the tenant.

SSPR: https://docs.microsoft.com/en-us/azure/active-directory/authentication/tutorial-enable-sspr

However, I need this screen to not appear in the first place. How do I achieve that?

The correct answer would be to actually register MFA if you want to follow the best practices.