The mentioned narrations are rejected by Shias and Sunnis, which suggests the lack of credibility in these traditions. Muslims were paying attention since the very early days of Islam in memorizing the Holy Book, reciting it, teaching it and spreading it. It was reported that the month of Ramadan is the spring season of the Holy Book, referring to the best time of the year. It was a common practice to recite the entire Quran during this month. Imam Mohamed Al-Baqir recited the Holy Book ten times during the holy month. It was also narrated that Imam Ali Al-Ridha did the same.
Since the beginning days of Islam, there were certain reciters who specialized in the recitation of the Quran and they taught it to others. As such, the Book was widespread amongst the Muslims in general, not to any special class in the Muslim society, and they went to great lengths to preserve the words of the Quran without any changes and alterations.
All of this indicates that the Holy Quran which we currently have, reached us from generations after generations. Therefore, the claim that you refer to in your question is meaningless and rejected.