There are of course a myriad of ways to go about it. As a DM, my standard game world for most of my gaming days was one wherein all classes and the standard PHB races were offered. Magic items were
extremely rare. Fighters were happy to find +1 swords, and were still using them at 8th plus levels. I always felt that once you handed away too much magic, it just got harder and harder to make the game fun and remotely challenging.
An extreme example of the opposite was a Rogue I played once. By the time he was 5th or 6th level, he had a ring that let him see in the dark like truesight up to 120 feet. The Cloak of the Bat, allowing him to turn invisible, and fly. Boots of Silence. Magic sword, Dagger of Throwing. A couple of rings of something or another. Amulet of something something The DM gave it away like candy it seemed. And I got bored
really fast. In the end, my thief skills didn't matter. All that mattered was the gear I collected. The fighter in our party didn't wield his sword. He let it dance and fight for him, a la the Dancing Sword. Our mage didn't need spells. He had a Staff of Fireballs, a Wand of Lightning, a Robe of the Archmage. Everything was easy. Again, this is an extreme example. After a couple of months of this, I was DMing again.
And that's the problem with magic. Sooner or later it just overshadows everything else. And players and DM's can believe that the classes are balanced, but they're not. At the later levels, and even earlier because of the overhaul that is 5e (I like it, don't get me wrong; it's simplicity is great), spellcasters are just exceedingly powerful when compared to non-casters. Just take a look through a universal spell list like
donjon. Sooner or later, it's clear as day: the non-casters
need the spellcasters. At the highest levels it's always coming down to magic, and it gets dull.
At this stage in my gaming preferences, I'm looking for something dark, grim. Not an evil world where everyone is a murder hobo, but one wherein skill and toughness matter more than some spellcaster casting Rope Trick and waiting for the monsters to get bored and wander off, confused.
Part of it goes with what Len said. The multiclassing will evolve into hybrid characters that are essentially a different class in their own right. I think you can keep the other races, but only if you dump everything that touches on magic: darkvision, hellsih rebuke, halfling luck rerolls, etc. Ultimately, I think it works better if you dump all the non-human races, and use different races within human, from the PHB. Also, to enrich a world that doesn't have magic, change character generation a bit. I'm all for point buy, but give everyone more points to spend. 6 extra would be the same as the +1 to all that already exists, but it gives the players a little more agency in spending the points. And let everyone start with two feats, as long as they fit the game world (e.g., not Mage Slayer, or Magic Initiate).
Dump all of the magic and magic classes, and magic archetypes, or anything that resembles magic. I think the DM, however, would have access to characters that can use magic, although they would be extremely rare, and powerful, likely archenemies in time, or key villains. Keep the creatures from the monster manual, but be smart as the DM and use the ones that the players can overcome with non-magical resourcefulness. Overcoming those challenges will be all the greater. Instead of wasting that vampire with fireballs and holy avenger swords, the players (wisely!) snuck the hell into its lair during daylight, overcame whatever it was to get to his coffin, and staked the bastard in the heart. And to do so, they 'held him down' when he awoke, in the coffin, with a holy symbol that's only purpose was to ward off evil. And the players had to go on a whole quest just for that.
Keep the rituals. Accessed only with the Ritual Caster feat. As the DM, choose the rituals that exist in your world. You control the access to what can even be cast. Create a simple home brew on how long it takes to cast them, rather than a generic 10 minutes. Change some spell levels. Maybe the level of rituals you can cast is equal to or less than your INT or WIS modifier. No matter how you do it, this keeps a touch of magic in your world, and keeps it more unique, rare, and special. Magic is more of a long lost art, a science in a way. Something that can be learned by anyone, with some practice. The key here is that nobody is dumping fireballs all the time, teleporting at the drop of hat, blinking around, shooting magic missiles, etc etc etc. If everything is a ritual, and you trim down what is available, heavily, magic is just way cooler. Suddenly you have a quest, to find a ritual, and it could be a Knock spell. The best part is this: it's a Knock spell that only works on one door, but you need it.
And players that start with the Ritual Caster feat, well, the DM can have a short list of more 'common' rituals that those practiced in the arts have passed down. Rituals practitioners might be few and far between, but they are out there. Now your backstory just has to make sense. Spells like Guidance, Produce Flame, Bless, Protection from Evil, etc. If it cold work as a ritual, call it a ritual. Just don't get silly with it.
And maybe those characters might find the occasional rare bit of magic as an item. The entire world doesn't have to be non-magical, just make it
rare, wherein even the simplest magic items are a wonder and a mystery. In D&D, when the players get a ring of invisibility, it's great, but nothing amazing. In Lord of the Rings it is practically a one of a kind item, and there aren't Wizards and Arcane Trickster Rogues everywhere going "Pfft. Big deal. I can do that at 3rd level."
D&D is great, but the magic just gets silly. And to each their own. For me, I want some tension in my adventure, some real drama.
