Let us start from a different question:
Why three books?
We could have just one; lots of games do. But that book often, though not always*, divides into two sections: "stuff for everyone" and "stuff for whoever runs the game." So that leads to a natural division of two books: a "Player's Guide" and a "Gamerunner's Guide". This is a reasonably common direction for "inspired by D&D, but not actually D&D" games, as well as just games-in-general, since "this is for general players" and "this isn't for general players, but for someone organizing things" is fairly straightforward.
But that still leaves us lacking an explanation for the third book. Obviously that's been the tradition for D&D, but it could have been otherwise. Why three books and not two? Simply put, it's because D&D wanted to have lots and lots and lots and lots of pre-written creatures to fight, and having to share page-space with the stuff telling the DM how to DM would be too much for a single volume.
An important lesson from this is that D&D has always put a high priority on the stats...and, in particular, the combat stats...of creatures. It wants to have nitty-gritty details about many, many different creatures. That's why we get Monster Manuals--to give us monsters, and in particular, monsters to slay. Even though the early editions of D&D treated combat like a failure state, it was still a sufficiently important thing that most creatures got stats you'd need if you picked a fight with them. Other than morale rules though, they rarely got much in the way of stats for other ways to encounter and interact with them--because that just wasn't enough of a priority to flesh out mechanically.
That, then, leads us to the first proper candidate for a "fourth" core book: A book about all the things adventurers do that aren't tangling with monsters. We have a Player's Handbook and a Monster Manual. Why not an Adventure Album? A thing that covers all the other aspects of Adventure: puzzle-solving, exploration, intrigue, faith, diplomacy, etc.
But there's another side, one that has never quite settled well and often gets a much-later supplement. Something every D&D game references and makes use of, but which is often overlooked or, annoyingly, dismissed as a minor concern. That is, equipment and magic items. Usually this, like monster stats in the two-book system, gets lumped with the DMG--but there have been times that it gets put in the player-focused book instead, and even 5e splits this with some stuff in the PHB and some in the DMG. An Equipment Emporium book also makes a lot of sense as a "fourth" core book, since every edition has had equipment matter (much as some might like to disparage modern editions for not doing so), and most groups like having magic items.
Two candidates, both plausible. I'm not sure there's a simple answer here. Maybe one book can do both jobs--after all, adventure is often about equipment!--but maybe not. Rationally though, something in that space of "the stuff we do and/or use while adventuring" seems the correct choice for a fourth core book. It's not any more "necessary" than three or even two core books, but I could see it. Hell, I could see an alternate reality where Gygax published four books originally, and now we'd be debating what a "fifth" core book would look like.
*Some games, like Ironsworn, don't need a GM and thus don't divide this way! Further, even games that do have GM role still only publish one core book. Most White Wolf games have one core book and several subgroup-specific supplements.
