Short answer
Yes. The Qur'an was always first a book to be recited. Its very name means "the recitation". The Prophet (ﷺ) taught it aloud. His companions learned it by heart and taught their students the same way. That chain has never broken. Today a student recites the whole Qur'an to a teacher. He then gets a certificate, called an ijaza. It names his teachers in a line back to the Prophet (ﷺ). The reading most Muslims use, Hafs, goes back through Asim of Kufa to Uthman, Ali and Ibn Mas'ud. In Ramadan, many mosques recite the whole Qur'an from memory. Those praying behind correct any slip. If every printed copy were lost, it could be written out again from memory.
Full explanation
A book meant to be recited
The word qur'an means recitation. The Qur'an says it was sent down in parts so that the Prophet (ﷺ) would recite it to people at intervals. It tells him that gathering it and reciting it is Allah's (﷾) work. It calls itself clear signs in the hearts of those given knowledge. William Graham, a Harvard historian of religion, describes the Qur'an as above all a spoken book: learned by heart and heard aloud, more than read in silence.
“(It is) a Qur'an which We have divided (into parts from time to time), in order that thou mightest recite it to men at intervals: We have revealed it by stages.”
“Move not thy tongue concerning the (Qur'an) to make haste therewith. It is for Us to collect it and to promulgate it: But when We have promulgated it, follow thou its recital (as promulgated): Nay more, it is for Us to explain it (and make it clear):”
“Nay, here are Signs self-evident in the hearts of those endowed with knowledge: and none but the unjust reject Our Signs.”
A Harvard historian of religion studies how scriptures live in speech and memory. In the part titled "An Arabic Reciting: Qur'an as Spoken Book", he describes the Qur'an as above all a recited scripture, whose name means recitation, learned by heart and heard aloud more than read silently.
Taught from mouth to ear
Every year the angel Gabriel went over the whole Qur'an with the Prophet (ﷺ), and twice in his last year. The Prophet (ﷺ) named four companions to learn it from: Ibn Mas'ud, Salim, Ubayy ibn Ka'b and Mu'adh ibn Jabal. He said the best of people are those who learn the Qur'an and teach it. The man who passed that saying on, Abu Abd al-Rahman al-Sulami, taught the Qur'an in Kufa for decades, from the rule of Uthman to the time of al-Hajjaj. He said that saying was what kept him in his teaching seat.
Abu Hurayra reports that Gabriel went over the Qur'an with the Prophet (ﷺ) once every year, and twice in the year he died.
Abdullah ibn Amr reports that the Prophet (ﷺ) named four men from whom the recitation of the Qur'an should be learnt: Abdullah ibn Mas'ud, Salim the freed slave of Abu Hudhayfa, Ubayy ibn Ka'b and Mu'adh ibn Jabal. Al-Bukhari places it in the chapter on the merits of Salim.
Uthman reports that the Prophet (ﷺ) said: the best of you are those who learn the Qur'an and teach it. The narrator adds that Abu Abd al-Rahman al-Sulami, who passed on this saying from Uthman, taught the Qur'an from the rule of Uthman until the time of al-Hajjaj, and said: that is what has kept me sitting in this seat of mine.
A chain anyone can follow
Ibn al-Jazari, a scholar of the readings, set out the chains of each reading in about 1400. The reading of Hafs, the one most Muslims recite today, is a clear example. Hafs learned it from Asim of Kufa. Asim learned from al-Sulami and others, who learned from Ibn Mas'ud, Uthman, Ali, Ubayy and Zayd ibn Thabit. They learned it from the Prophet (ﷺ). Ibn al-Jazari lists dozens of separate routes by which that one reading reached him. Teachers still pass on the Qur'an this way, and a student who completes it is given an ijaza: a certificate naming his teachers in a line back to the Prophet (ﷺ).
Ibn al-Jazari sets out the chains of the reading of Asim of Kufa (d. 745 CE), which Hafs passed on. Hafs learned it from Asim. Asim learned from Abu Abd al-Rahman al-Sulami, Zirr ibn Hubaysh and Abu Amr al-Shaybani, who learned from Abdullah ibn Mas'ud. Al-Sulami and Zirr also learned from Uthman and Ali, and al-Sulami from Ubayy ibn Ka'b and Zayd ibn Thabit. These companions learned it from the Prophet (ﷺ). Asim led the teaching of the Qur'an in Kufa after al-Sulami. Ibn al-Jazari lists dozens of routes by which the reading reached him.
The Prophet (ﷺ)
He recited it, and taught it aloud.
His Companions
Ibn Mas'ud, Uthman, Ali, Ubayy and Zayd ibn Thabit.
Their students
Al-Sulami and others, in Kufa.
Asim of Kufa
The master whose reading Hafs passed on.
Hafs
The reading most Muslims recite today.
A student today
Recites the whole Qur'an to a teacher and receives an ijaza naming the chain back to the Prophet (ﷺ).
Memory first, paper second
Ibn al-Jazari wrote that in passing on the Qur'an the reliance is on what is kept in hearts, not on written copies. He called this the noblest gift Allah (﷾) gave this community. He pointed to a hadith in Sahih Muslim. In it Allah (﷾) tells the Prophet (ﷺ): I have sent down to you a Book which water cannot wash away, which you recite asleep and awake. A book held in many memories does not depend on any one copy.
Ibn al-Jazari, the master of the Qur'an's readings, writes that in passing on the Qur'an the reliance is on what is kept in hearts and memories, not on written copies and books, and calls this the noblest gift Allah gave this community. He cites the hadith in Sahih Muslim of a Book which water cannot wash away.
In a long sermon, Iyad ibn Himar reports, the Prophet (ﷺ) told of what Allah had said to him, including: I have sent down to you a Book which water cannot wash away, which you recite asleep and awake.
Checked aloud, in public
In Ramadan, Umar gathered the people for the night prayer behind one reciter, Ubayy ibn Ka'b. Mosques around the world still recite the Qur'an in these night prayers, and many finish the whole of it in the month. The imam often recites from memory, and anyone behind him who knows it will call out if he slips. A text recited aloud before crowds, every year, in every land, cannot drift without someone noticing. The Qur'an says Allah (﷾) has made it easy to remember.
Abd al-Rahman ibn Abd al-Qari reports that Umar found people in the mosque in Ramadan praying the night prayer in scattered groups. He gathered them behind one reciter, Ubayy ibn Ka'b. Later, seeing them pray together, Umar said, "What an excellent new thing (bid'a) this is". He added that the prayer at the end of the night, which they slept through, was better.
“And We have indeed made the Qur'an easy to understand and remember: then is there any that will receive admonition?”
Two witnesses that agree
Memory and writing kept the Qur'an side by side from the start. When many who knew it by heart were killed at the battle of Yamama, Abu Bakr had it gathered into one written volume, from written pieces and from the memories of men. The written copy was a safeguard for memory, not a replacement. The two can still be checked against each other. The oldest leaves, kept in Birmingham and dated to between 568 and 645 CE, carry the same words that are recited from memory today.
Zayd ibn Thabit reports that after many who knew the Qur'an by heart were killed at Yamama, Abu Bakr asked him to gather the Qur'an into one volume, reminding him that he used to write down the revelation for the Messenger of Allah. Zayd collected it from written pieces and from the memories of men, and the finished copy passed from Abu Bakr to Umar and then to Hafsa.
Two parchment leaves carrying parts of Surahs 18 to 20 in early Hijazi script. In 2015 the Oxford Radiocarbon Accelerator Unit dated the parchment to between 568 and 645 CE, with 95.4 per cent probability, a span that covers the Prophet's (ﷺ) lifetime (c. 570-632). Apart from small points of spelling, the text on the leaves is the text of those surahs as the Qur'an has them today.
Islam's answer: guarded as promised
The Qur'an says Allah (﷾) sent down the Message and will guard it. It says no falsehood can approach it from before or behind. Allah (﷾) kept that promise through people: generation after generation who learned it by heart, taught it face to face, and checked one another aloud.
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
“No falsehood can approach it from before or behind it: It is sent down by One Full of Wisdom, Worthy of all Praise.”
The replies answered
Sahih al-Bukhari 4986 — Kitab Fada'il al-Qur'an (Virtues of the Qur'an), hadith 4986
Zayd ibn Thabit reports that after many who knew the Qur'an by heart were killed at Yamama, Abu Bakr asked him to gather the Qur'an into one volume, reminding him that he used to write down the revelation for the Messenger of Allah. Zayd collected it from written pieces and from the memories of men, and the finished copy passed from Abu Bakr to Umar and then to Hafsa.
Cited for
If memory was enough, why did Abu Bakr need to gather the Qur'an in writing?
The response
Because many who knew it by heart had just been killed at Yamama, and Umar feared more would die. The written volume was a safeguard for memory, gathered from writing and from the men who knew it. The companions had not written one volume earlier because the Qur'an lived in their prayers and memories.
Where it lands — Writing was added to protect memory, not to replace it.
Perspectives
Memory changes what it passes on
Some sceptics say that anything passed on by word of mouth changes over time, like a message whispered down a line. So a text kept by memory cannot be trusted.
Experiments show that a story passed from person to person grows shorter and changes with each retelling.
Rebuttal
Those stories were heard once and retold freely. The Qur'an is learned word for word, recited aloud in every prayer, checked by teachers and by crowds, and kept beside written copies. The oldest leaves agree with what is recited today.
Muslims recite the Qur'an in more than one reading, so memory has produced different versions.
Rebuttal
The readings were not created by faulty memory. The Prophet (ﷺ) himself taught the Qur'an in more than one mode, and each accepted reading has its own chain back to him. None contradicts another.
The Islamic perspective
The Qur'an was taught aloud and learned by heart from the start, and it is still passed on face to face, with named chains back to the Prophet (ﷺ). Memory and writing confirm each other.
The Qur'an was revealed in parts to be recited to people.
The reading of Hafs reaches the Prophet (ﷺ) through Asim, al-Sulami and companions such as Uthman, Ali and Ibn Mas'ud.
The Prophet (ﷺ) said the best of people are those who learn the Qur'an and teach it.
Allah (﷾) promised to guard the Message He sent down.