This is very easy to prove that it is untrue. We not only have several copies of the codex, but also in the last century we found the Dead Sea scrolls from the qumran community, which confirmed that the scriptures in most of the bible matched what was written 2,000 years ago.

One thing to understand is that there was no real council for the selection of the scriptures. The Church used the Septuagint (Greek Old Testament Scriptures), which was pretty much the only copies of the scriptures they would have that the majority of people could read. And when it came for the New Testament, we don't have things that were outright destroyed, we have things that the community of Christians rejected. Take for instance the Gospels of Thomas, Judas, and Mary. Each were written a few centuries later than the 4 gospels included in the Canon of the New Testament, More than that, the things taught in them were contrary to what people knew of the character of Christ (in Thomas, they had the Christ child killing people). BUT we still have copies of those texts, they weren't destroyed, they just didn't include them. You can also tell because of how many copies of the scriptures there are compared to how many copies there are of the other texts.

But there are also books that aren't considered scripture (but at one point were included in the readings of the Church), such as the Protoevangelium of James, the Shepherd of Hermas, the martyrdom of Saint Polycarp. They are considered good for reading, but not reliable enough to have been included in Scripture.

At last, a question I can answer!

I can't speak about popes, but at least one religious leader definitely edited the Bible to suit his own sensibilities: Hong Xiuquan, leader of the 19th-century Taiping movement in China.

Background

The Taipings were a quasi-Christian insurrectionist movement that seized several major cities and large swathes of territory in southern China during the 1850s and 1860s before losing their capital of Nanjing in 1864. About 20 million people [sic] died as a result of the war.

Their leader, Hong Xiuquan, had a series of visions in which he learned that he was the younger brother of Jesus Christ. Hong and a handful of early supporters, especially Hong Rengan (yes, related), had received a few Christian tracts from British and American missionaries, and had even been baptized.

When the Taipings, who called themselves 拜上帝教, "the God-Worshipping Religion," were harassed by Qing forces, they fought back. They met with some success, and their ideology quickly took on violent anti-Qing tones. It should be noted that anti-Qing sentiment was common in China, not least for ethnic reasons. The Qing were Manchu, a non-Sinitic people whom the majority Han considered barbarians. Throughout the Qing dynasty there were attempts to overthrow it and restore Han rule. This period was no exception, and during the Taiping insurrection the Qing were dealing with several other concurrent rebellions elsewhere in China.

Editing the Bible

Having taken Nanjing, the Taipings undertook a massive printing project involving 600 craftsmen. The goals: (a) register every household in the city, and (b) spread religious literature to everyone. Here's Spence [emphasis added]:

The first book of the Bible that the Taiping publish is the Book of Genesis, chapters 1 through 28, from the creation of the world through Jacob's dream of the ladder... Even though the Chinese translation is not that fluid or perfect--at times not even clear--the Taiping block carvers are instructed to print all twenty-eight chapters directly from Gutzlaff's text, just as they find them, not excluding the numbers of each verse that are inserted among the columns. There is only one exception, something the Taiping leaders find so shameful that they cannot put it in their people's hands. That is the last eight verses of Genesis 19...

That's the part where Lot's daughters get him drunk and commit incest with him. That crosses a line, so Hong orders it excised. Conveniently, the verse numbering doesn't have to be adjusted because the story comes at the end of the chapter.

That may have been the only change to the first part of Genesis, but Hong eventually ordered other changes too. For example, references to Jesus as God's only son were changed to accommodate Hong's own status as Jesus' younger brother.

But much more drastic views on the Bible were coming.

Forget the Bible, It's the Yang Show Now

After taking Nanjing, Hong Xiuquan retreated into the inner palace and hung out there for the remainder of the war, occasionally issuing decrees but leaving operational control to others. One major leader was Yang Xiuqing, dubbed "East King." Yang claimed the ability to speak directly for God the Father. The following incident with the British led the Taipings, at the direction of Yang, to downplay the Bible altogether in favour of direct revelation from Yang and other approved sources.

In the summer of 1854, the British ship Rattler on a diplomatic mission made contact with the Taipings in Nanjing. Two observations about the subsequent written exchange: (a) It showed how differently the Taipings viewed the Christian deity--namely, they viewed God as unmistakably corporeal; and (b) Yang emphasized, in no uncertain terms, that Hong Xiuquan was the literal, physical brother of Jesus and that Yang himself was divinely appointed. The British disputed both.

Regarding corporeality, here's a sample of questions from Yang to the British:

You nations having worshipped God for so long a time, does any one among you know:

1. How tall God is, or how broad?

2. What his appearance or colour is?

3. How large his abdomen is?

Other questions concern God's attire and other children, and then repeat the same questions about Jesus. The British responded by denying the corporeality of God, and saying that God had no son but Jesus.

British response regarding Hong's relationship to Jesus, and the divine appointment of Yang:

In reference to your closing declaration, such as that God has specially commissioned you and your people to exterminate the imps [anti-Taiping forces, with a spritz of genocide--/u/envatted_love]--that your sovereign is God's own son, and the uterine brother of the Celestial Elder Brother--that he is the true sovereign of all nations--that you, the Eastern King, are appointed by God to the office of the Holy Ghost, the Comforter--I think it right to state to you distinctly that we place no faith in any one of your dogmas to this effect... We believe only what is revealed to us in the Old and New Testaments, namely that God the Father is the creator and Lord of all things--that Jesus is his only begotten son...

A week later, Yang has an announcement to make (speaking as God the Father) to all the other Taiping officials:

Your God has come down to you today for one reason and one reason only: namely to inform you that both the Old Testament and the New Testament, which have been preserved in foreign lands, contain numerous falsehoods. You are to inform the North King and the the Wing King, who in turn will tell the East King who can inform the Heavenly King, that it is no longer useful to propagate these books.

From that point, the Taipings no longer had to work to reconcile their beliefs with the Bible. Note that Yang died in a failed coup in 1856, two years later.

Source: God's Chinese Son, Jonathan Spence (1996)

(My copy is on Kindle, so if you want page numbers someone will have to tell me how to get those from "locations.")