Randall J Stephens' 2018 book The Devil's Music: How Christians Inspired, Condemned, and Embraced Rock'n'Roll has a chapter specifically devoted to the 'bigger than Jesus' controversy, which is useful in terms of putting it in context in terms of the way that American conservative Christians reacted to pop culture in the 1960s.

Firstly, it's important to remember that the Supreme Governor of the Church of England - The Queen - awarded MBEs to the Beatles in 1965, which at the time was sort of seen as an apology for taxing their earnings so heavily. Judging by a 2021 article by Marcus Collins in Popular Music, The Beatles certainly were criticised by religious conservatives in the UK - Malcolm Muggeridge (later famous for a televised debate with Michael Palin and John Cleese over Life Of Brian) argued that the Beatles phenomenon was essentially 'the bad dreams of a materialistic society', while the English Catholic Christopher Booker thought that Beatlemania was indicative of the declining psychic health of England, brought on by declining class distinctions and decolonisation.

According to Stevens, Lennon's comments to Maureen Cleave in an April 1966 edition of the London broadsheet newspaper the Evening Standard (later republished in a magazine aimed at teen girls in the US) were based on his having read the 1965 book The Passover Plot by Hugh J. Schonfield, a bestseller which argued that the Passion narrative in the gospels was a sort of conspiracy/plot that was carefully designed to make Jesus look like the messiah, and that Jesus's disciples were largely unaware of the machinations behind events.

In the UK context, anyone reading Lennon's comments would probably made the connection that Lennon had read The Passover Plot. Lennon's takeaway from The Passover Plot was, if you read his comments in full, that the same social hysteria that he saw in The Passover Plot was present in the Passion narrative was also present in 1960s Beatlemania; Lennon was deeply uncomfortable with the common phenomenon of disabled children being brought backstage to be touched by a Beatle, a discomfort he expressed through impersonating/parodying disabled people on stage (which, let's face it, is Lennon being an ableist asshole). Lennon felt that these children were being brought to them to be healed by a pop band, and he hated it. So the remarks passed without much comment, as Lennon was contributing to a discourse in British society that already existed.

In terms of why it provoked a massive uproar in the US, the Cleave interview came to be published in the U.S. not in a similar broadsheet newspaper, but instead in the teen-girl oriented Datebook. In a very 'won't somebody think of the children?!' kind of way, it's one thing to discuss the Beatles being more popular than Jesus in a hoity-toity newspaper that will largely be read by sober adults, but it's another thing to tell impressionable teen girls that Christianity will become extinct.

But also, if you look at the reaction to the 'bigger than Jesus' comment in the US, it's noticeably dominated by the Bible Belt, that part of the US where evangelical Protestantism has retained its influence (and which very often had a history of slavery and then Jim Crow - which Paul McCartney actually attacked in an interview republished in that same issue of Datebook). To give a flavour of the most disturbing reactions to the 'bigger than Jesus' comment, from the Stephens book:

The South Carolina grand dragon of the KKK, for instance, burned a cross with a Beatles record affixed to it. A Klavern in Mississippi sponsored a Beatles wig burning and accused the group of “atheistic” leanings. In Memphis one Klansman described Lennon’s comment as “nothing but blasphemy.” “And we’re gonna try to stop it,” he stated menacingly, even if it meant the use of “terror.” Imperial wizard Robert Shelton served up what he likely thought would be the ultimate insult. The former Tuscaloosa tire salesman told a British reporter, “It’s hard for me to tell through the mop-heads . . . whether they’re even white or black.”

The geographical location of the radio stations that banned the Beatles from their airwaves also tells a similar story: Stephens mentions WRNB in New Bern, North Carolina, WAKY in Louisville, Kentucky, KOB in Albuquerque, New Mexico, WAQY in Birmingham, Alabama, two stations in Texas, and one station in South Dakota.

Broadly speaking, Southern evangelical Protestantism was hostile to American secular pop culture in the first place. As the 1960s progressed and the 1960s counterculture that the Beatles became increasingly attached to became more prominent in the media, elements of Southern evangelical Protestantism saw the encroachment of moral deterioration. Amongst other things, in 1960, a Catholic was elected as President; in 1962, the Supreme Court banned official school prayer; and in 1963, the reading of scripture in public schools was forbidden.

Stephens quotes a Protestant pastor who said in 1965 that:

My sympathy is very much with young people these days. They're getting filth thrown at them from every conceivable source.

Infamously, in April 1966, Time magazine had published a controversial cover article titled, 'Is God Dead?', echoing Nietzsche's sentiments from almost a century beforehand (as the title of an article about Christian thinkers trying to find new ways to move forward in an increasing secular world). Lennon's comments in the US, then, were taken by socially conservative Southern evangelicals as proof in the pudding that the Beatles were part of this moral deterioration.

The conservative Christians behind the Beatles boycott were already concerned by Beatlemania, which was of course such a prominent feature of pop culture in 1964-1965 that they could not have failed to notice it. One Dr. Bernard Saibel said in the Boston Globe of the Beatles' concerts that:

The hysteria and loss of control go far beyond the impact of the music...[young people] became frantic, hostile, uncontrolled, screaming, unrecognizable beings...[normal polite] girls behaved as if possessed by some demonic urge, defying in emotional ecstasy the restraints which authorities try to place on them...[overall it was] unholy bedlam.

Conservative Christians initially thought that the Beatles were a fad, like the fad around Elvis, that would soon pass. Comments by Billy Graham, soon after watching an Ed Sullivan Show performance, reflected this general belief; the Beatles "represent the restlessness and the longing of young people today for something off-beat, something dif­ferent" (which Graham of course saw as "confusion", as that 'something different' should in his eyes be Jesus).

However, as Beatlemania continued, concern about it grew. According to Stephens:

Writing to Playboy magazine, of all places, in March 1965, a San Francisco father worried that his daughter was taking Beatles worship to new, ridiculous heights. “It may seem sort of silly, but things have reached the stage where” even this reader of the titillating bachelor magazine was “getting worried.” He related how his “daughter and a number of the other kids in the neighborhood have formed a real cult over the Beatles.” The enthralled teens had “built an altar in one girl’s bedroom and they burn candles and recite Beatle prayers they have written.” They even went so far as to compose “a Beatle Bible which starts out, ‘in the beginning the Beatles created rock and roll.’"

This concern was also present in fundamentalist preachers:

Why had so many youngsters fallen for Beatlism? asked a well-known fundamentalist revival preacher. Such stalwarts turned to scripture for answers. It was simple, he figured. “The devil has control of this world right now.” Moreover, “the Devil knows that if teenagers and adults hear the ‘glorious gospel of Christ’ that they will be saved and begin to live for Christ.” Accordingly, Lucifer had blinded youths and turned their attention to the Beatles.

As a result, fundamentalist preachers encouraging Beatles fans to burn their Beatles records was already happening in 1965, well before Lennon made his comments. According to Stephens:

In 1965, the Bakkers ended their large teen evangelism campaign with a “Beatle Burning.” They encouraged local youth to toss “objectionable books and magazines” and rock ’n’ roll records into the flames.

In contrast to the evangelicals, 'mainline Protestants' in the US were less likely to want to forbid the Beatles, and saw the controversy as an opportunity for soul-searching. Stephens reports that:

Writing for Presbyterian magazine, William Stringfellow even ventured that the “Beatles’ premise is correct” in some measure. True, he wrote, the “churches in Great Britain and, increasingly, the churches in North America endure significant public disfavor, I venture, because they are so much preoccupied with themselves and so little attentive to the common needs of human beings in modern society.”

But, basically, Lennon's comments were indicative of a struggle between secular society and religious society.

In truth, the rock star’s comment, however flippant, was certainly in line with the prevailing wisdom of critics, opinion makers, and academics. Throughout the 1960s, numerous observers were sure that organized religion, even in America, was on the decline. The uproar that surrounded the Time magazine God-is-dead story was only one manifestation of that consensus. Secularism was the order of the day. Those who denied it, said sociologists and philosophers, were deluding themselves. Secularization theory served as the master model of the era.