The internet didn't exist in 1964 but if it had existed, the Beatles would still have been big, and Motown would still have been big sellers. For example, even in 1964 Motown artists were performing on TV, and people obviously knew they were black. But they still bought the records, still supported those artists.
The internet has transformed the way news and other media are consumed, but it hasn't actually sped up the transmission of news all that much. It's interactive, and that's different, of course, and the massive amount of information beats what people got in 1964 from TV, radio, Newspapers and magazines.
But TV and radio back then weren't exactly Pony Express. And anything notable usually was mentioned on radio, TV, or in the newspapers.
When Kennedy was shot in November 1963, everyone knew because of TV and radio. When he died, everyone knew. And when I say "everyone", yeah, not literally every person in the country had a radio or TV nearby, but word got out just the same -- that word being transmitted via radio and TV.
The smartphone has replaced the radio, TV and newspaper, but it doesn't mean the populace is any better informed.
RE: Lennon's 'Jesus' statement -- the Beatles almost did become dead meat because of it. There were Beatle record burnings in some areas of the US. Lennon's retraction statement helped quell the negative reactions, but even at the time he made the original statement there were many in the media who understood what Lennon actually meant -- his original point being expounded in later interviews: that the Beatles were popular on a level even Lennon couldn't comprehend, and to a level he thought was insane. He told of one woman rolling up a son in a wheelchair before or after a meet and greet, hoping Lennon could heal him. Lennon freaked out over this.
At the same time, there have been some who make equally volatile statements in the internet age and they haven't become dead meat because of it. There is so much going on that is covered by the internet, that people gradually become inurred to such statements, compared to maybe 1920 or 1960. Today, people are overloaded with info.