I myself was always one who preferred invulnerable weapons, weapons that won't break no matter how much you use them because sometimes there's a weapon I really like and I want to keep it. That said, invulnerable weapons aren't a real thing and it would stand to reason that some games would have weapon durability for the sake of immersion and realism.
Before I get into this I want to stress that while I will bring it up quite a bit, I do not want The Legend of Zelda: Breath of the Wild to be the main focus of this post. Naturally, it brought weapon durability to the forefront of people's thinking because of how critically and commercially successful it has become but this is far from the first game ever to feature it. Games like Dark Souls, Dark Cloud, and many others that came before Breath of the Wild have featured weapon durability so I do not want to place the sole responsibility for its existence and discussion on it.
Now, I'm of two minds on this particular topic: on the one hand, I see the point of view of the supporters of this design philosophy. Breakable weapons does have some advantages in terms of some combat systems. However, I also see the other side of the argument, that weapon durability is not all that worth while of a mechanic so I want to go into detail about the ins and outs of this kind of mechanic in relation to gear based combat systems.
For those who don't know, a gear based combat system is a combat system that relies heavily on the gear you have currently equipped. Unlike skill based combat systems, which reward proficiency and dexterity and punish laziness and bitching out on easy modes, gear based combat systems have a much greater reliance on the gear you currently have and how effective it will be against your next big set of opponents.
Now, let's try to look at the positives of this kind of system. First of all, this kind of combat system rewards pre-planning and preparation. If the game is well built around this, the events leading up to your next big encounter will give you an idea of what gear will be best suited to the fight so that you can properly find that gear and use it. For example, in the first area of Demon's Souls the boss Phalanx is vulnerable to fire and subtly tells you throughout your journey to find it that the fire bombs are a good piece of equipment to have for the area. In this way, you can either fight Phalanx normally with a basic sword and shield or you can use the fire bombs and lighten up your load.
Of course, Phalanx is the first boss of the game anyway and is arguably not really a difficult one. So instead let's take a look at Dark Cloud. Dark Cloud has a crafting system to go along with its weapon sets and you consistently have to repair your weapons using repair powder. In a particular fight in the second area of the game, you have to get a particular sword in order to progress because the serpent that's blocking your path is invulnerable toward everything else except that sword. Once you get the sword, you can progress and you realize the game gave you a fairly powerful weapon to deal with enemies with. It made sure that you were ready for what was to come in the rest of the area and also made sure that you know how to craft weapons, something that some players may not have gotten even if most had.
This kind of combat system also rewards exploration by making it necessary to find certain items you need for a given fight so that you can be equipped to deal with it. Naturally, in a gear based combat system, it stands to reason that most of your gear is going to fall out of use eventually, either because later gear is stronger or because of weapon durability. In The Legend of Zelda Breath of the Wild, if you go to the South at the beginning of the game you will find a really powerful centaur enemy. This one, unlike other enemies in the game, does not carry around its neck the weapon you need to defeat it. It also has a lot of hearts and high durability. Meaning that, if you go straight away, even if you fill every weapon slot with some weapon you found in the beginning stage, it is likely that all your weapons will break before it dies, assuming you don't get killed first.
In this way, the game is punishing you because you came unprepared. If you went to a different area and gathered some stronger weapons you might've been able to kill it. This incentivizes you to go look for stronger weapons so you don't run into that situation again.
And thirdly, it makes balancing less of a chore. In a game with permanent weapons that you can hold forever, all of the weapons you use have to be fairly balanced against each other. They may have their strengths and weaknesses that make them useful in some situations over other weapons, for example in DMC3 Agni & Rudra is best suited toward crowd control whereas Beowulf is best suited to one on one engagements, in general every weapon should be suitable for any given fight in the entire game.
If a game has breakable weapons though, this is not much of an incentive since the weapon's probably going to break before the end of your second or third fight with it so even if it's the worst weapon in the game, that's okay because you'll find better stuff. This also more or less balances out your power as well, since even if you have the best weapons in the game, they'll break as well so you're not too overpowered for too long.
Now that we've looked at the upsides to this, what are the downsides? Well, there are quite a few. The first and most obvious to someone like me is that having a breakable weapon means that you're not as likely to have a weapon you want when you need it. In a game like Devil May Cry, Ninja Gaiden, Bayonetta, or Metal Gear Rising: Revengeance, having your favorite weapon is a fairly big chunk of the gameplay time since you're likely to continue playing with whatever of weapons you're best with or what you find most enjoyable.
This reliance on gathering new gear to replace the old ones means that you can't really build depth into the combat system. In most high tier action games, the depth comes from what actions you can perform and how useful those actions are. In a game that is much more gear based, you can't really have any number of differing attacks because the weapons are going to break long before you have the chance to master any of the movesets. Instead, numbers are far more important since, if it's going to break, you may as well equip whatever's strongest and kill the toughest enemies with it.
On the developer's side, this lack of implementing differing moves may also be somewhat related to having to create all different kinds of weapon models that can be used to replace broken weapons, since it takes some time and money away from the animators who would be animating the diverse movesets.
Rather than adding depth, the weapon durability actually subtracts depth by removing options and forcing you to find new ones. Since depth comes from the meaningful decisions you can make during a fight, the gear as the focus of the combat system makes the choice simply of what to equip and how long to use it.
In a game like Dark Souls, this is less of an issue. While weapon durability is a mechanic in that game, the only things that chip away at it significantly are the toxin status and certain collision glitches so, for the most part, finding a weapon you need is not as much of an issue. Dark Cloud does this less well but the Broken Dagger Glitch makes it so that you can save money on upgrades and instead buy repair powder to keep your current weapon from breaking. It's not much but it is something.
In Breath of Wild, however, the breakable weapons break easily enough that constant replacements are needed, which in turn disincentivizes you from growing attached to any given weapon. That said, while Breath of the Wild is bad in this regard, Way of the Samurai may actually be worse. Way of the Samurai also has breakable weapons but also implements different and complex movesets for the weapons you can equip. While this does add an air of wonder to the weapons you can gather from defeated opponents, since powerful opponents tend to have really cool swords, the weapon durability takes something away from it, since it could break a weapon you were just starting to get attached to.
In this situation, two mechanics are at odds with each other: the weapon durability, which promotes finding replacements for broken weapons, and the complex movelists, which incentivize you to master a weapon to get really good at using it.
Needless to say these mechanics don't really mesh well which is why it's important to think about how your mechanics work with each other. By adding weapon durability, it lessens the need for skillful play by making it about how strong your equipment is and how well it helps you. Without weapon durability, the weapons you want to use are dependent on how well they accommodate your current skill level. There are merits to both and while I do prefer the latter myself, I will say that it's possible to like both, neither, one or the other.