Yes, each member of the expanded affiliated group (EAG) still needs to separately register even if the lead foreign financial institution (FI) registers and lists each member on Part 2 of the registration.
The retrieved information specifies that each member financial institution of an EAG must have a registration completed. While the lead FI can create the FATCA accounts for its members and provide them with the necessary FATCA IDs and temporary access codes, each member FI must log into the system and complete its registration. Alternatively, the lead FI can complete the registration on behalf of the member FIs. This ensures that each member FI has its own account and is properly registered under FATCA requirements.
Sources:
Instructions for Form 8957 (06/2018)
Publication 5118 (07/2021)
§ 1.1471-4. FFI agreement.