It's basically hard to trust charts before the mid-1950s or so to be an accurate record of what we now want to see when looking at charts, because the people putting together those charts just simply didn't think in terms of sales of individual recordings by a particular artist.

Instead, the music industry's metrics were structured around songs. So if you look at old issues of Billboard before the 1950s, the charts were song charts, not charts of the sales of a particular single sung by a particular artist put out on a particular record company. So, for example, the Hank Williams song 'Cold Cold Heart' was #2 on the 'Best Sellers in Stores' list in 1951, but that #2 was for the song itself, not for Hank Williams' recording, or anybody else's recording (though from context, I'd say that Tony Bennett's recording was actually the most popular of the recordings of the song). So Whitburn's book you mention, which essentially attempts to understand what more modern Billboard charts would look like if they had existed before the mid-1950s, is essentially based on a bunch of guesswork; but Whitburn has also written several other chart books based on more modern data which are official Billboard publications, and so his guesswork is at least official guesswork done by someone who likely knows the ins and outs of the composition of the modern Billboard chart (something which is a trade secret).

I should also mention that the way that Billboard does the charts tries to get a sense of what songs are most popular amongst a range of different metrics, from sales and radio airplay to jukebox spins (back in the day) to (more recently) streaming data. Billboard's Hot 100 chart basically tries to identify the songs that feel the most ...now. In contrast, until quite recently, the standard charts in Australia and the UK are entirely based on sales (more recently they've integrated streaming data given the decline of the iTunes Music Store and the rise of streaming services). So - in terms of #1s, what exactly makes a #1 is up for grabs, and different people argue for different metrics, especially for time spans when the current definition of a #1 in a particular territory doesn't apply.

So Bing Crosby having more #1s than the Beatles is simply something we cannot definitively state; the Beatles come from a time when there were #1s, and when having a #1 had a certain status - the #1, before the mid-1950s, was not something that had status outside of the music industry. Even by December 1957, on an early episode of American Bandstand, they don't appear to do a countdown of the Top 10 of the week; Top Of The Pops (in which the whole point is the popularity of the songs, and in which the countdown is integral) was first broadcast in 1964, and by about then American Bandstand was doing a Top 10 countdown as part of the show. But before this, people just seem like they didn't care what recording was nationally the most popular (individual radio stations would have charts of their own which claimed to say what was popular within in the area they broadcast to, based on requests, or sales in affiliated stores).

The Australian equivalent of Joel Whitburn is David Kent, who has compiled a chart book that goes back to 1940, which I own; Kent's method of finding out what the #1 was before 1966 (when the first national Australian chart appeared in press, with 'Yellow Submarine' by the Beatles at #1) is to basically look at a bunch of these radio station published regional charts and collate them using a similar principle to how they calculated it from 1966 onwards (which was based on collating and weighting those regional charts, at least into the 1970s). But before the late 1950s, if you look at Kent's charts, it's the song rather than the particular recording which was on the chart.

So, to talk about it in terms of Bing Crosby vs the Beatles, Bing Crosby was definitely incredibly popular, incredibly influential, and incredibly lucrative for the record companies he signed to. He invented and popularised a popular music singing style, more or less, which was the dominant style for a good three decades. His version of 'White Christmas' sometimes gets called the world's best selling single ever. The Beatles were also definitely incredibly popular, in a way that simply hadn't existed previously (but which was aided by a bunch of record industry changes Bing Crosby didn't have access to, such as the rise of the hit parade culturally). But Bing Crosby had been a recording artist for a good decade before Billboard first published a 'hit parade' style chart (in 1936), let alone something resembling the modern Hot 100. And secondly, the charts that he did appear on after 1936 fundamentally weren't about the popularity of Bing Crosby singing a song, they were just about the popularity of a song. Fairly comparing them is apples and oranges, really.