Google Classroom is generally managed by the School's IT. Your setup is strange if they request public gmail account and not internal account in the school tenant.
- Your teacher is correct in not being happy. Giving their personal email to a student is... problematic
- Remember that your privacy is already breached when a 3rd party in the communication is using gmail
- Simply create a burner gmail account and do not use that for anything else than inside your school's google ecosystem