botw is a game about exploring and scrounging, and totk even moreso, and one of the big things you scrounge for is weapons. and like half of the game is built around this. if weapons didn't break readily then you'd just carry around Good Sword until you found Gooder Sword and then just carry that around instead
(incidentally, one of the reasons i actively like weapon durability is that i had this as a specific complaint about many previous zelda games — the item upgrade curve has this awkward pacing where you get the coolest stuff at the end of the game where you won't really have anywhere left to use it, but if you manage to get cool stuff earlier then the whole game is a breeze because now you have it forever)
breakable weapons mean no weapon is forever, so you can be excited more than once when you run across something unusual or quirky — i will, after all, always want another fire sword. breakable weapons mean you want to carry spares, so there's a compelling reason to need more inventory space, so there's a point to finding koroks. breakable weapons mean that it's mechanically interesting that enemies drop their own weapons. breakable weapons give you a reason to revisit areas where you know good weapons are available, rather than the typical rpg thing where you go through part of the world and then it's basically empty lifeless. breakable weapons turn melee combat into one of several viable combat options with a clear drawback, rather than just the obvious thing you do by default every time
in a less well-designed game, it would be very bad to feel like i'm constantly on the verge of becoming helpless. but Sharp Things are abundant, Good Sharp Things are only a little bit out of the way once you learn where they are, and there are generally still half a dozen other things you could do besides charge in waving a sword around. weapon durability is a very clever bridge between exploration and combat that makes the two feed neatly into one another, and that's not an easy feat
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