Is a full recitation of the Quran counted for each participant in a collective reading of parts of it, and does the reward differ based on the intention?

This method is not a full recitation (khatmah) of the Qur'an, and we fear it may be an innovation (bid'ah) because it was not part of the guidance of the early Muslims (Salaf), and it does not achieve any sharia-compliant purpose.