Why do teeth don't regenerate?

You said exactly why in your post: "...our biological design..."

There's no such thing. We evolved. That means we're a mix of traits passed along over time by individuals that managed to live long enough to breed.

That's it. That's the whole explanation for any question about "why don't humans do x thing as part of our biology?"

Any given trait is all about lasting long enough to make babies. Once that occurs, all that's left is a general proclivity to ensuring the babies survive long enough to do the same. Regrowing teeth isn't part of that. It's a niche trait that isn't as useful as you'd think for humans. We don't need to gnaw at things, we don't need to crack bones with our mouths, nothing that would make a third set of teeth an advantage, or different teeth an advantage.

Teeth are not easily breakable. We actually can crack bone with our jaws and the teeth will usually survive if the bone isn't too thick; we just have better tools for that because way back when, the proto-humans that used tools had more babies that survived to make more babies. You have to abuse and/or neglect your teeth to break them for the vast majority. There are congenital issues where that isn't the case, but we've also bred ourselves into a social species that takes care of each other, so we aren't limited to a harsh, primitive survival level of things.

I really don't get why people think of teeth as fragile. They're incredibly durable for what we need them for, and require only minimal care to last well beyond breeding age. Even if you factor in modern diets being bad for teeth, regular care for them (brushing and flossing) can stave off those effects for decades. Go search up some of the dental research on old human bodies from archaeological sites. People survived very well with just one set of adult teeth.

And, some humans do have extras that can come in later in life, though it's very rare and comes with drawbacks (according to the last lady I dated that was an anthropologist anyway). Supposedly, having the extras actually weakens the regular adult teeth and makes them more prone to damage. There's always a tradeoff in things like this.