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 Muhammad Al-Baqir (peace be upon him) recited the Holy Book ten times during the holy month. It was also narrated that Imam Ali Al-Ridha (peace be upon him) did the same.
Since the earliest 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 generation after generation. Therefore, the claim that you refer to in your question is meaningless and rejected.

The sequence of the verses is same as the sequence in the time of the Messenger of Allah (peace be upon him and his holy progeny).

The current sequence of the verses is the same as the sequence at the time of the Prophet (peace be upon him and his progeny).

We are not certain of the sequence of the verses’ revelations. We are sure of the fact that the Quran is the current contemporary one.

There is no doubt that Almighty Allah protected the Holy Quran from any changes or alterations and such protection played an important role in guarding the religion and protecting it from alterations and changes. The reason for that might be the fact that the verses of the Holy Quran were collected in one book and guarded during the lifetime of the Prophet (peace be upon him) and continued to be like that generation after generation until today.

His Eminence answered a group of questions that were raised to him from a Sunni researcher in which he said the following regarding the completion of the Quran: “Muslims from Shia and Sunni sects have unanimously agreed by practical means that whatever between the two covers in the current book is the complete Holy Quran. The current books of Quran that are spread in all Muslim countries serve as evidence to this truth.
Muslims consider reciting the entirety of the current copies of the Quran to be the complete reciting of the Book of Allah. Similarly, a recitation of a Surah in the current Quran is considered to be the recitation of the complete Surah. They do not add a word or remove another. The said is clear in their understanding, conduct and in the verdicts of their religious scholars.
This practical consensus is one of the strongest proofs of the Muslims’ position in general – whether they are Shia or Sunni – on the Holy Quran. Nothing can stand against such evidence as it expresses the fact that the noble Quran is entirely extant in the current book as an existing and self-evident truth. Muslims have acknowledged it and implemented it, while all fallacies seeking to cause doubt in it have failed.”