First, this assumption is wrong:

keyboards are very common in rock bands, especially today -- but they aren't part of the "standard" three- or four-piece set, and were less common at rock's outset.

Though the term "rock and roll" wasn't popularized until ~1955, the musical style had been around for a decade, or more, by then. There's a very fine line between mid-50s "rock and roll" and late-40s "jump blues" and "rhythm and blues".

In the late 40s/early 50s, the genre we now know as rock and roll often featured piano as its centerpiece. The names Fats Domino and Little Richard are pretty familiar to rock and roll aficionados -- less so are their original contemporaries like Joe Liggins, Professor Longhair, and Piano Red, all of whom were piano players in the rock and roll style.

There were also the bands fronted by "blues shouters" such as Big Joe Turner, Wynonie Harris, and H-Bomb Ferguson. Their ensembles were usually fronted by piano and/or saxaphone, as typified by Big Joe Turner's 1954 recording of "Shake, Rattle, and Roll". To most ears, it's unmistakably rock and roll, but doesn't contain electric guitar at all. Perhaps more convincingly, Little Richard and Jerry Lee Lewis records almost never featured guitar, yet they are universally considered pioneers of the rock and roll genre. Similarly, many early rock groups like Bill Haley & His Comets would feature rhythm guitar but the solos would be played by the saxophone player.

There were two important record labels in the mid-50s that played the most integral part in making the electric guitar the centerpiece of the emerging genre: Chess Records in Chicago and Sun Records in Memphis. Chess famously made stars of black guitarists Chuck Berry, Bo Diddley, and Muddy Waters. (Waters is often dismissed as a "blues musician" though even from his 1947 Chess Records debut, his upbeat electric guitar/drums/bass ensemble laid the groundwork for rock and roll.)

Sun Records brought us many of the most important early white rock and rollers such as Elvis Presley, Jerry Lee Lewis, Carl Perkins, and (before he became associated with country) Johnny Cash.

What you'll notice about all these artists is that virtually none of them used the "standard" guitar/guitar/bass/drums lineup often associated with rock. Bo Diddley had a five-piece band that played with him into the mid-60s, the lineup being guitar/drums/bass/piano/maracas. Chuck Berry's was either that same lineup, or else paired down to just guitar/drums/bass, like Muddy Waters'.

Likewise, at Sun Records, many of the early rock groups were trios. In his earliest days, Elvis played rhythm guitar backed by lead guitarist Scotty Moore and upright bassist Bill Black, as can be heard on their first single "That's All Right (Mama)". Johnny Cash's backing band was called "The Tennesse Two" -- again, just bass and second guitar, without a drummer.

Even outside those labels, non-four piece bands were common, as typified by Buddy Holly & The Crickets (originally a three piece), Johnny Burnette & The Rock & Roll Trio (another three piece), Gene Vincent (a five-piece), and so on.

So, back to your original question: how did guitar/guitar/bass/drums become the "standard"? Well, it never really has been the standard. The simplest way to look at it is that it's the lineup that Elvis used from ~1955 until the end of the 1950s, and perhaps most importantly, it's the lineup the Beatles used (more or less) throughout their recording career. Certainly their live career post-1961. Though even then, a four-piece band wasn't anything near a strict requirement. There were plenty of British Invasion bands that had five members such as the Rolling Stones, the Dave Clark Five, the Yardbirds, and the Zombies. And in their aftermath, the "power trio" quickly became a thing, as popularized by the Jimi Hendrix Experience, Cream, and Blue Cheer.

In other words, there really is no "standard" rock and roll lineup, nor has there ever been. If you do see a "standard", then you're probably looking through the lens of the Beatles and Elvis, i.e. the two most successful rock and roll acts of the 20th Century. That they were both four-piece bands was more or less coincidence. The Beatles were still a five-piece band only a year or so before they began their recording career, and Elvis was recording as a three-piece while recording his first singles. Before then and after then, rock and roll lineups have never been rigid, and have featured a variety of instruments to play both rhythm and lead parts.

I think the question should be "Why did the guitar become the standard instrument for modern rock music?"

Modern rock music is very much a guitar-based genre - we can add the bass guitar as a kind of guitar as well.

Guitars and drums are the lowest common denominator in all of rock's modern sub-genres to the point where rock band and guitar band are almost synonymous. "Rock" as opposed to "rock 'n roll" became heavier, more guitar-based, riff-based, louder, more distorted and became something very different from rock 'n roll to the point where new music that one would've once called "rock 'n roll" would be now called country, blues or something else.

It really is the guitar's popularity, more than anything, that seemingly created a genre of itself - a genre of bands with 3 or more guitars. The guitar also became a major instrument in jazz, country and bluegrass to name a few genres.

In a way, we shifted our definition of what "rock" is to something much heavier (in music terms) and much more guitar-based than what rock 'n roll originally was. Our image of what a rock band or rock star is shifted to one that is "guitar band" or "guitar player" and images of a rock star smashing a guitar are more "rock" to the average person than someone playing the piano or the saxophone.

We not only shifted the definition, but reduced its scope to exclude things that not only would've been called rock 'n roll before, but to exclude someone once called "the king of rock 'n roll" as well.

TL;DR we have shifted our definition of rock music to be heavier and reduced its scope to become synonymous with guitar band. Guitars are the lowest common denominator in guitar bands, from thrash like Megadeth to grunge like Nirvana, progressive blues rock like Pink Floyd and so on.

We also love creating sub-genres and bands love using sub-genres to define themselves so bands with different instruments use that to stand out and not just be a "Rock band" but "Cello Rock" or "Psychobilly" etc.

The guitar ended up being more versatile then those other instruments. It could be used to play both chords or single line melodies. Keyboards like pianos are usually based on chords. It was light and portable unlike keyboards of the past. Minimal skills were needed to start, just learn to shape three chords and you can play a song. Also, the early blues guitarists figured out if they began to alter the tunings of the guitar it meant a player could create completely new chords with minimal effort. Adding electric to guitars changed it even more with amplifiers giving not just massive increases in volume but also brand new sounds especially the tube amps of Fender and later Marshall. Finally distortion pedals allowed even more creativity.